QuickSort Parosh Aziz Abdulla Uppsala University September 12, 2012 Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 1/7
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QuickSort
Parosh Aziz Abdulla
Uppsala University
September 12, 2012
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 1 / 7
1 The Partition Procedure
2 Algorithm
3 Worst-Case Behavior
4 Best-Case Behavior
5 Randomized QuickSort
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 2 / 7
The Partition Procedure
Partition
Partition(A, p, r)
1 x ← A[r ]2 i ← p − 13 for j ← p to r − 14 do if A[j] ≤ x5 then i ← i + 16 exchange A[i ]↔ A[j]7 exchange A[i + 1]↔ A[r ]8 return i + 1
2 9 8 1 3 6 7 5
i
jp r
2 9 8 1 3 6 7 5
ip j r
2 9 8 1 3 6 7 5
ip j r
2 9 8 1 3 6 7 5
ip j r
2 1 8 9 3 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i r
2 1 3 5 8 6 7 9
p i r
Partition(A, 1, 8)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 3 / 7
The Partition Procedure
Partition
Partition(A, p, r)
1 x ← A[r ]2 i ← p − 13 for j ← p to r − 14 do if A[j] ≤ x5 then i ← i + 16 exchange A[i ]↔ A[j]7 exchange A[i + 1]↔ A[r ]8 return i + 1
2 9 8 1 3 6 7 5
i
jp r
2 9 8 1 3 6 7 5
ip j r
2 9 8 1 3 6 7 5
ip j r
2 9 8 1 3 6 7 5
ip j r
2 1 8 9 3 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i r
2 1 3 5 8 6 7 9
p i r
Partition(A, 1, 8)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 3 / 7
The Partition Procedure
Partition
Partition(A, p, r)
1 x ← A[r ]2 i ← p − 13 for j ← p to r − 14 do if A[j] ≤ x5 then i ← i + 16 exchange A[i ]↔ A[j]7 exchange A[i + 1]↔ A[r ]8 return i + 1
2 9 8 1 3 6 7 5
i
jp r
2 9 8 1 3 6 7 5
ip j r
2 9 8 1 3 6 7 5
ip j r
2 9 8 1 3 6 7 5
ip j r
2 1 8 9 3 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i r
2 1 3 5 8 6 7 9
p i r
Partition(A, 1, 8)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 3 / 7
The Partition Procedure
Partition
Partition(A, p, r)
1 x ← A[r ]2 i ← p − 13 for j ← p to r − 14 do if A[j] ≤ x5 then i ← i + 16 exchange A[i ]↔ A[j]7 exchange A[i + 1]↔ A[r ]8 return i + 1
2 9 8 1 3 6 7 5
i
jp r
2 9 8 1 3 6 7 5
ip j r
2 9 8 1 3 6 7 5
ip j r
2 9 8 1 3 6 7 5
ip j r
2 1 8 9 3 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i r
2 1 3 5 8 6 7 9
p i r
Partition(A, 1, 8)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 3 / 7
The Partition Procedure
Partition
Partition(A, p, r)
1 x ← A[r ]2 i ← p − 13 for j ← p to r − 14 do if A[j] ≤ x5 then i ← i + 16 exchange A[i ]↔ A[j]7 exchange A[i + 1]↔ A[r ]8 return i + 1
2 9 8 1 3 6 7 5
i
jp r
2 9 8 1 3 6 7 5
ip j r
2 9 8 1 3 6 7 5
ip j r
2 9 8 1 3 6 7 5
ip j r
2 1 8 9 3 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i r
2 1 3 5 8 6 7 9
p i r
Partition(A, 1, 8)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 3 / 7
The Partition Procedure
Partition
Partition(A, p, r)
1 x ← A[r ]2 i ← p − 13 for j ← p to r − 14 do if A[j] ≤ x5 then i ← i + 16 exchange A[i ]↔ A[j]7 exchange A[i + 1]↔ A[r ]8 return i + 1
2 9 8 1 3 6 7 5
i
jp r
2 9 8 1 3 6 7 5
ip j r
2 9 8 1 3 6 7 5
ip j r
2 9 8 1 3 6 7 5
ip j r
2 1 8 9 3 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i r
2 1 3 5 8 6 7 9
p i r
Partition(A, 1, 8)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 3 / 7
The Partition Procedure
Partition
Partition(A, p, r)
1 x ← A[r ]2 i ← p − 13 for j ← p to r − 14 do if A[j] ≤ x5 then i ← i + 16 exchange A[i ]↔ A[j]7 exchange A[i + 1]↔ A[r ]8 return i + 1
2 9 8 1 3 6 7 5
i
jp r
2 9 8 1 3 6 7 5
ip j r
2 9 8 1 3 6 7 5
ip j r
2 9 8 1 3 6 7 5
ip j r
2 1 8 9 3 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i r
2 1 3 5 8 6 7 9
p i r
Partition(A, 1, 8)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 3 / 7
The Partition Procedure
Partition
Partition(A, p, r)
1 x ← A[r ]2 i ← p − 13 for j ← p to r − 14 do if A[j] ≤ x5 then i ← i + 16 exchange A[i ]↔ A[j]7 exchange A[i + 1]↔ A[r ]8 return i + 1
2 9 8 1 3 6 7 5
i
jp r
2 9 8 1 3 6 7 5
ip j r
2 9 8 1 3 6 7 5
ip j r
2 9 8 1 3 6 7 5
ip j r
2 1 8 9 3 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i r
2 1 3 5 8 6 7 9
p i r
Partition(A, 1, 8)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 3 / 7
The Partition Procedure
Partition
Partition(A, p, r)
1 x ← A[r ]2 i ← p − 13 for j ← p to r − 14 do if A[j] ≤ x5 then i ← i + 16 exchange A[i ]↔ A[j]7 exchange A[i + 1]↔ A[r ]8 return i + 1
2 9 8 1 3 6 7 5
i
jp r
2 9 8 1 3 6 7 5
ip j r
2 9 8 1 3 6 7 5
ip j r
2 9 8 1 3 6 7 5
ip j r
2 1 8 9 3 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i j r
2 1 3 9 8 6 7 5
p i r
2 1 3 5 8 6 7 9
p i r
Partition(A, 1, 8)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 3 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)(A,1,10)(A,1,10)
(A,1,3)(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,10)(A,5,10)
(A,5,6)(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)(A,1,10)(A,1,10)
(A,1,3)(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,10)(A,5,10)
(A,5,6)(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)
QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,10)(A,5,10)
(A,5,6)(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)
QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)
Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,10)(A,5,10)
(A,5,6)(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)
QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)
Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,10)(A,5,10)
(A,5,6)(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
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9
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9
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9
8
9
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9
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9
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9
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9
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9
4
10
4
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6
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7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)
QuickSort(A, 1, 3)
QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
3
3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
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8
5
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2
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2
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5
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9
5
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9
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9
8
9
8
9
4
10
4
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6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)
QuickSort(A, 1, 3)
QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)
Partition(A, 1, 3)
Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
3
3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
5
8
5
8
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8
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6
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2
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9
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9
4
10
4
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6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)
QuickSort(A, 1, 3)
QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)
Partition(A, 1, 3)
Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
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3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
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4
4
4
4
8
5
8
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2
8
2
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5
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9
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9
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9
8
9
4
10
4
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6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
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2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
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8
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8
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9
8
9
4
10
4
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6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)
QuickSort(A, 1, 1)
QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
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3
2
3
2
3
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3
3
3
3
3
3
3
6
4
6
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4
4
4
4
8
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9
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9
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6
10
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10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
3
3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
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8
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8
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2
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2
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9
8
9
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10
4
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6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)
QuickSort(A, 3, 3)
QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
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3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
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8
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8
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6
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2
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9
8
9
4
10
4
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6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
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3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
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8
5
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8
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6
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6
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7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)
QuickSort(A, 5, 10)
QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
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10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)
QuickSort(A, 5, 10)
QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)
Partition(A, 5, 10)
Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
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6
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8
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10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)
QuickSort(A, 5, 10)
QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)
Partition(A, 5, 10)
Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
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9
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9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
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9
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9
10
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9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)
QuickSort(A, 5, 6)
QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
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9
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9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)
QuickSort(A, 5, 6)
QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)
Partition(A, 5, 6)
Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
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10
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10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)
QuickSort(A, 5, 6)
QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)
Partition(A, 5, 6)
Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)(A,5,4) (A,6,6)(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
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6
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10
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10
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10
9
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9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)
(A,5,4)
(A,6,6)
(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
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6
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9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)
QuickSort(A, 5, 4)
QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)
(A,5,4)
(A,6,6)
(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
3
3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
5
8
5
8
5
8
5
6
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2
8
2
8
9
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8
5
9
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9
5
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
4
10
4
10
6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)
(A,5,4)
(A,6,6)
(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
3
3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
5
8
5
8
5
8
5
6
5
6
5
6
5
5
5
5
5
6
6
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6
6
6
6
6
6
6
5
6
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5
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6
6
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7
7
7
6
7
6
7
2
8
2
8
9
8
9
8
9
8
9
8
9
8
9
8
7
8
7
8
5
9
5
9
5
9
5
9
5
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
4
10
4
10
6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)
QuickSort(A, 6, 6)
QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)
(A,5,4)
(A,6,6)
(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
3
3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
5
8
5
8
5
8
5
6
5
6
5
6
5
5
5
5
5
6
6
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6
6
6
6
6
6
5
6
5
6
5
6
6
6
6
6
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6
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7
7
7
7
7
7
7
6
7
6
7
2
8
2
8
9
8
9
8
9
8
9
8
9
8
9
8
7
8
7
8
5
9
5
9
5
9
5
9
5
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
4
10
4
10
6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)
(A,5,4) (A,6,6)
(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
3
3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
5
8
5
8
5
8
5
6
5
6
5
6
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5
5
5
5
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7
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7
6
7
2
8
2
8
9
8
9
8
9
8
9
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9
8
9
8
7
8
7
8
5
9
5
9
5
9
5
9
5
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
4
10
4
10
6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)
QuickSort(A, 8, 10)
QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)
(A,5,4) (A,6,6)
(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
3
3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
5
8
5
8
5
8
5
6
5
6
5
6
5
5
5
5
5
6
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6
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6
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6
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7
7
6
7
6
7
2
8
2
8
9
8
9
8
9
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9
8
9
8
9
8
7
8
7
8
5
9
5
9
5
9
5
9
5
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
4
10
4
10
6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)
QuickSort(A, 8, 10)
QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)
Partition(A, 8, 10)
Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)
(A,5,4) (A,6,6)
(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
3
3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
5
8
5
8
5
8
5
6
5
6
5
6
5
5
5
5
5
6
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5
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6
7
6
7
2
8
2
8
9
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9
8
9
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9
8
9
8
9
8
7
8
7
8
5
9
5
9
5
9
5
9
5
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
4
10
4
10
6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)
QuickSort(A, 8, 10)
QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)
Partition(A, 8, 10)
Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)
(A,5,4) (A,6,6)
(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)(A,8,7) (A,9,10)(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
3
3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
5
8
5
8
5
8
5
6
5
6
5
6
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
5
6
5
6
5
6
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6
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6
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6
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7
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7
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6
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6
7
2
8
2
8
9
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9
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9
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9
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9
8
9
8
7
8
7
8
5
9
5
9
5
9
5
9
5
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
4
10
4
10
6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)
(A,5,4) (A,6,6)
(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)
(A,8,7)
(A,9,10)
(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
3
3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
5
8
5
8
5
8
5
6
5
6
5
6
5
5
5
5
5
6
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6
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6
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7
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7
6
7
2
8
2
8
9
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9
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9
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9
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8
7
8
5
9
5
9
5
9
5
9
5
9
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9
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9
8
9
8
9
8
9
8
9
8
9
8
9
4
10
4
10
6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)
QuickSort(A, 8, 7)
QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)
(A,5,4) (A,6,6)
(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)
(A,8,7)
(A,9,10)
(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
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3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
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8
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2
8
2
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5
9
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9
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9
5
9
5
9
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9
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9
8
9
8
9
8
9
8
9
8
9
8
9
4
10
4
10
6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)
(A,5,4) (A,6,6)
(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)
(A,8,7)
(A,9,10)
(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
3
3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
5
8
5
8
5
8
5
6
5
6
5
6
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
5
6
5
6
5
6
6
6
6
6
6
6
7
7
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7
7
7
7
7
7
7
6
7
6
7
2
8
2
8
9
8
9
8
9
8
9
8
9
8
9
8
7
8
7
8
5
9
5
9
5
9
5
9
5
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
4
10
4
10
6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)
QuickSort(A, 9, 10)
QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)
(A,5,4) (A,6,6)
(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)
(A,8,7)
(A,9,10)
(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
3
3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
5
8
5
8
5
8
5
6
5
6
5
6
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
5
6
5
6
5
6
6
6
6
6
6
6
7
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7
7
7
7
7
7
7
7
6
7
6
7
2
8
2
8
9
8
9
8
9
8
9
8
9
8
9
8
7
8
7
8
5
9
5
9
5
9
5
9
5
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
4
10
4
10
6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)
QuickSort(A, 9, 10)
QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)
Partition(A, 9, 10)(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)
(A,5,4) (A,6,6)
(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)
(A,8,7)
(A,9,10)
(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
3
3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
5
8
5
8
5
8
5
6
5
6
5
6
5
5
5
5
5
6
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6
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6
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7
7
6
7
6
7
2
8
2
8
9
8
9
8
9
8
9
8
9
8
9
8
7
8
7
8
5
9
5
9
5
9
5
9
5
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
4
10
4
10
6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)
QuickSort(A, 9, 10)
QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)
Partition(A, 9, 10)(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)
(A,5,4) (A,6,6)
(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)
(A,8,7)
(A,9,10)
(A,9,10)
(A,9,9)(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
3
3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
5
8
5
8
5
8
5
6
5
6
5
6
5
5
5
5
5
6
6
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6
6
6
6
6
5
6
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6
5
6
6
6
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6
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7
7
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7
7
7
6
7
6
7
2
8
2
8
9
8
9
8
9
8
9
8
9
8
9
8
7
8
7
8
5
9
5
9
5
9
5
9
5
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
4
10
4
10
6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)
(A,5,4) (A,6,6)
(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)
(A,8,7)
(A,9,10)
(A,9,10)
(A,9,9)
(A,9,9)
(A,11,10)
(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
3
3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
5
8
5
8
5
8
5
6
5
6
5
6
5
5
5
5
5
6
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6
6
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6
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6
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5
6
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6
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7
7
7
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7
6
7
2
8
2
8
9
8
9
8
9
8
9
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9
8
9
8
7
8
7
8
5
9
5
9
5
9
5
9
5
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
4
10
4
10
6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)
QuickSort(A, 9, 9)
QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)
(A,5,4) (A,6,6)
(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)
(A,8,7)
(A,9,10)
(A,9,10)
(A,9,9)
(A,9,9)
(A,11,10)
(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
3
3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
5
8
5
8
5
8
5
6
5
6
5
6
5
5
5
5
5
6
6
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6
6
6
6
6
6
6
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6
5
6
5
6
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6
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6
6
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7
7
7
7
7
7
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6
7
6
7
2
8
2
8
9
8
9
8
9
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9
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9
8
9
8
7
8
7
8
5
9
5
9
5
9
5
9
5
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
4
10
4
10
6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)
(A,5,4) (A,6,6)
(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)
(A,8,7)
(A,9,10)
(A,9,10)
(A,9,9)
(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
3
3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
5
8
5
8
5
8
5
6
5
6
5
6
5
5
5
5
5
6
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6
6
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7
6
7
2
8
2
8
9
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9
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9
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9
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9
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9
8
7
8
7
8
5
9
5
9
5
9
5
9
5
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
4
10
4
10
6
10
6
10
6
10
7
10
7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)
QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)
(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)
(A,5,4) (A,6,6)
(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)
(A,8,7)
(A,9,10)
(A,9,10)
(A,9,9)
(A,9,9) (A,11,10)(A,11,10)
9
1
9
1
1
1
1
1
1
1
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8
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9
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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Algorithm
Quick Sort
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 10)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 10)QuickSort(A, 5, 6)QuickSort(A, 5, 4)QuickSort(A, 6, 6)QuickSort(A, 8, 10)QuickSort(A, 8, 7)QuickSort(A, 9, 10)QuickSort(A, 9, 9)QuickSort(A, 11, 10)
Partition(A, 1, 10)Partition(A, 1, 3)Partition(A, 5, 10)Partition(A, 5, 6)Partition(A, 8, 10)Partition(A, 9, 10)(A,1,10)
(A,1,10)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,10)
(A,5,10)
(A,5,6)
(A,5,6)
(A,5,4)
(A,5,4) (A,6,6)
(A,6,6)
(A,8,10)
(A,8,10)
(A,8,7)
(A,8,7) (A,9,10)
(A,9,10)
(A,9,9)
(A,9,9) (A,11,10)
(A,11,10)
9
1
9
1
1
1
1
1
1
1
1
1
1
2
1
2
3
2
3
2
3
2
2
2
2
2
3
3
3
3
2
3
2
3
2
3
3
3
3
3
3
3
6
4
6
4
4
4
4
4
8
5
8
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8
5
8
5
8
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6
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6
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2
8
2
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9
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9
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9
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8
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8
5
9
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9
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9
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9
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9
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9
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8
9
4
10
4
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10
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7
10
7
10
9
10
9
10
9
10
9
10
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 4 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)(A,1,8)
(A,1,7)(A,1,7) (A,9,8)(A,9,8)
(A,1,6)(A,1,6) (A,8,7)(A,8,7)
(A,1,5)(A,1,5) (A,7,6)(A,7,6)
(A,1,4)(A,1,4) (A,6,5)(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
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1
1
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1
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3
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4
4
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7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)(A,1,8)
(A,1,7)(A,1,7) (A,9,8)(A,9,8)
(A,1,6)(A,1,6) (A,8,7)(A,8,7)
(A,1,5)(A,1,5) (A,7,6)(A,7,6)
(A,1,4)(A,1,4) (A,6,5)(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
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7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)(A,1,7) (A,9,8)(A,9,8)
(A,1,6)(A,1,6) (A,8,7)(A,8,7)
(A,1,5)(A,1,5) (A,7,6)(A,7,6)
(A,1,4)(A,1,4) (A,6,5)(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
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8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)
QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)(A,1,7) (A,9,8)(A,9,8)
(A,1,6)(A,1,6) (A,8,7)(A,8,7)
(A,1,5)(A,1,5) (A,7,6)(A,7,6)
(A,1,4)(A,1,4) (A,6,5)(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
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8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)
QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)
Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)(A,1,7) (A,9,8)(A,9,8)
(A,1,6)(A,1,6) (A,8,7)(A,8,7)
(A,1,5)(A,1,5) (A,7,6)(A,7,6)
(A,1,4)(A,1,4) (A,6,5)(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
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Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)
QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)
Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)(A,1,7) (A,9,8)(A,9,8)
(A,1,6)(A,1,6) (A,8,7)(A,8,7)
(A,1,5)(A,1,5) (A,7,6)(A,7,6)
(A,1,4)(A,1,4) (A,6,5)(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
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8
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1
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8
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1
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8
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1
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8
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1
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1
1
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8
8
1
1
2
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3
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7
8
8
1
1
2
2
3
3
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5
6
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7
8
8
1
1
2
2
3
3
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5
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7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)(A,1,6) (A,8,7)(A,8,7)
(A,1,5)(A,1,5) (A,7,6)(A,7,6)
(A,1,4)(A,1,4) (A,6,5)(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
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5
5
6
6
7
7
8
8
1
1
2
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3
3
4
4
5
5
6
6
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7
8
8
1
1
2
2
3
3
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5
5
6
6
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7
8
8
1
1
2
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3
3
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5
5
6
6
7
7
8
8
1
1
2
2
3
3
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5
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7
8
8
1
1
2
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3
3
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5
6
6
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7
8
8
1
1
2
2
3
3
4
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5
5
6
6
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7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
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7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)
QuickSort(A, 1, 7)
QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)(A,1,6) (A,8,7)(A,8,7)
(A,1,5)(A,1,5) (A,7,6)(A,7,6)
(A,1,4)(A,1,4) (A,6,5)(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
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5
5
6
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7
8
8
1
1
2
2
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3
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5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
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5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)
QuickSort(A, 1, 7)
QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)
Partition(A, 1, 7)
Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)(A,1,6) (A,8,7)(A,8,7)
(A,1,5)(A,1,5) (A,7,6)(A,7,6)
(A,1,4)(A,1,4) (A,6,5)(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)
QuickSort(A, 1, 7)
QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)
Partition(A, 1, 7)
Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)(A,1,6) (A,8,7)(A,8,7)
(A,1,5)(A,1,5) (A,7,6)(A,7,6)
(A,1,4)(A,1,4) (A,6,5)(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)(A,1,5) (A,7,6)(A,7,6)
(A,1,4)(A,1,4) (A,6,5)(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)
QuickSort(A, 1, 6)
QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)(A,1,5) (A,7,6)(A,7,6)
(A,1,4)(A,1,4) (A,6,5)(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)
QuickSort(A, 1, 6)
QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)
Partition(A, 1, 6)
Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)(A,1,5) (A,7,6)(A,7,6)
(A,1,4)(A,1,4) (A,6,5)(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)
QuickSort(A, 1, 6)
QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)
Partition(A, 1, 6)
Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)(A,1,5) (A,7,6)(A,7,6)
(A,1,4)(A,1,4) (A,6,5)(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)(A,1,4) (A,6,5)(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)
QuickSort(A, 1, 5)
QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)(A,1,4) (A,6,5)(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)
QuickSort(A, 1, 5)
QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)
Partition(A, 1, 5)
Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)(A,1,4) (A,6,5)(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)
QuickSort(A, 1, 5)
QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)
Partition(A, 1, 5)
Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)(A,1,4) (A,6,5)(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)
QuickSort(A, 1, 4)
QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)
QuickSort(A, 1, 4)
QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)
Partition(A, 1, 4)
Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)
QuickSort(A, 1, 4)
QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)
Partition(A, 1, 4)
Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)(A,1,3) (A,5,4)(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)
(A,1,3) (A,5,4)
(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)
QuickSort(A, 1, 3)
QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)
(A,1,3) (A,5,4)
(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)
QuickSort(A, 1, 3)
QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)
Partition(A, 1, 3)
Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)
(A,1,3) (A,5,4)
(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)
QuickSort(A, 1, 3)
QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)
Partition(A, 1, 3)
Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)
(A,1,3) (A,5,4)
(A,5,4)
(A,1,2)(A,1,2) (A,4,3)(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)
(A,1,3) (A,5,4)
(A,5,4)
(A,1,2)
(A,1,2) (A,4,3)
(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)
QuickSort(A, 1, 2)
QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)
(A,1,3) (A,5,4)
(A,5,4)
(A,1,2)
(A,1,2) (A,4,3)
(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)
QuickSort(A, 1, 2)
QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)
Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)
(A,1,3) (A,5,4)
(A,5,4)
(A,1,2)
(A,1,2) (A,4,3)
(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)
QuickSort(A, 1, 2)
QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)
Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)
(A,1,3) (A,5,4)
(A,5,4)
(A,1,2)
(A,1,2) (A,4,3)
(A,4,3)
(A,1,1)(A,1,1) (A,3,2)(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)
(A,1,3) (A,5,4)
(A,5,4)
(A,1,2)
(A,1,2) (A,4,3)
(A,4,3)
(A,1,1)
(A,1,1) (A,3,2)
(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)
QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)
(A,1,3) (A,5,4)
(A,5,4)
(A,1,2)
(A,1,2) (A,4,3)
(A,4,3)
(A,1,1)
(A,1,1) (A,3,2)
(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)
(A,1,3) (A,5,4)
(A,5,4)
(A,1,2)
(A,1,2) (A,4,3)
(A,4,3)
(A,1,1)
(A,1,1) (A,3,2)
(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)
QuickSort(A, 3, 2)
QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)
(A,1,3) (A,5,4)
(A,5,4)
(A,1,2)
(A,1,2) (A,4,3)
(A,4,3)
(A,1,1)
(A,1,1) (A,3,2)
(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)
(A,1,3) (A,5,4)
(A,5,4)
(A,1,2)
(A,1,2) (A,4,3)
(A,4,3)
(A,1,1)
(A,1,1)
(A,3,2)
(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)
QuickSort(A, 4, 3)
QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)
(A,1,3) (A,5,4)
(A,5,4)
(A,1,2)
(A,1,2) (A,4,3)
(A,4,3)
(A,1,1)
(A,1,1)
(A,3,2)
(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)
(A,1,3) (A,5,4)
(A,5,4)
(A,1,2)
(A,1,2)
(A,4,3)
(A,4,3)
(A,1,1)
(A,1,1)
(A,3,2)
(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)
QuickSort(A, 5, 4)
QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)
(A,1,3) (A,5,4)
(A,5,4)
(A,1,2)
(A,1,2)
(A,4,3)
(A,4,3)
(A,1,1)
(A,1,1)
(A,3,2)
(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)
(A,1,3)
(A,5,4)
(A,5,4)
(A,1,2)
(A,1,2)
(A,4,3)
(A,4,3)
(A,1,1)
(A,1,1)
(A,3,2)
(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)
QuickSort(A, 6, 5)
QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4) (A,6,5)
(A,6,5)
(A,1,3)
(A,1,3)
(A,5,4)
(A,5,4)
(A,1,2)
(A,1,2)
(A,4,3)
(A,4,3)
(A,1,1)
(A,1,1)
(A,3,2)
(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4)
(A,6,5)
(A,6,5)
(A,1,3)
(A,1,3)
(A,5,4)
(A,5,4)
(A,1,2)
(A,1,2)
(A,4,3)
(A,4,3)
(A,1,1)
(A,1,1)
(A,3,2)
(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)
QuickSort(A, 7, 6)
QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5) (A,7,6)
(A,7,6)
(A,1,4)
(A,1,4)
(A,6,5)
(A,6,5)
(A,1,3)
(A,1,3)
(A,5,4)
(A,5,4)
(A,1,2)
(A,1,2)
(A,4,3)
(A,4,3)
(A,1,1)
(A,1,1)
(A,3,2)
(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5)
(A,7,6)
(A,7,6)
(A,1,4)
(A,1,4)
(A,6,5)
(A,6,5)
(A,1,3)
(A,1,3)
(A,5,4)
(A,5,4)
(A,1,2)
(A,1,2)
(A,4,3)
(A,4,3)
(A,1,1)
(A,1,1)
(A,3,2)
(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)
QuickSort(A, 8, 7)
QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6) (A,8,7)
(A,8,7)
(A,1,5)
(A,1,5)
(A,7,6)
(A,7,6)
(A,1,4)
(A,1,4)
(A,6,5)
(A,6,5)
(A,1,3)
(A,1,3)
(A,5,4)
(A,5,4)
(A,1,2)
(A,1,2)
(A,4,3)
(A,4,3)
(A,1,1)
(A,1,1)
(A,3,2)
(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6)
(A,8,7)
(A,8,7)
(A,1,5)
(A,1,5)
(A,7,6)
(A,7,6)
(A,1,4)
(A,1,4)
(A,6,5)
(A,6,5)
(A,1,3)
(A,1,3)
(A,5,4)
(A,5,4)
(A,1,2)
(A,1,2)
(A,4,3)
(A,4,3)
(A,1,1)
(A,1,1)
(A,3,2)
(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)
QuickSort(A, 9, 8)
QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7) (A,9,8)
(A,9,8)
(A,1,6)
(A,1,6)
(A,8,7)
(A,8,7)
(A,1,5)
(A,1,5)
(A,7,6)
(A,7,6)
(A,1,4)
(A,1,4)
(A,6,5)
(A,6,5)
(A,1,3)
(A,1,3)
(A,5,4)
(A,5,4)
(A,1,2)
(A,1,2)
(A,4,3)
(A,4,3)
(A,1,1)
(A,1,1)
(A,3,2)
(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Worst-Case Behavior
QuickSort: Worst-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 8)QuickSort(A, 9, 8)QuickSort(A, 1, 7)QuickSort(A, 8, 7)QuickSort(A, 1, 6)QuickSort(A, 7, 6)QuickSort(A, 1, 5)QuickSort(A, 6, 5)QuickSort(A, 1, 4)QuickSort(A, 5, 4)QuickSort(A, 1, 3)QuickSort(A, 4, 3)QuickSort(A, 1, 2)QuickSort(A, 3, 2)QuickSort(A, 1, 1)
Partition(A, 1, 8)Partition(A, 1, 7)Partition(A, 1, 6)Partition(A, 1, 5)Partition(A, 1, 4)Partition(A, 1, 3)Partition(A, 1, 2)
(A,1,8)
(A,1,8)
(A,1,7)
(A,1,7)
(A,9,8)
(A,9,8)
(A,1,6)
(A,1,6)
(A,8,7)
(A,8,7)
(A,1,5)
(A,1,5)
(A,7,6)
(A,7,6)
(A,1,4)
(A,1,4)
(A,6,5)
(A,6,5)
(A,1,3)
(A,1,3)
(A,5,4)
(A,5,4)
(A,1,2)
(A,1,2)
(A,4,3)
(A,4,3)
(A,1,1)
(A,1,1)
(A,3,2)
(A,3,2)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 5 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)(A,1,15)
(A,1,7)(A,1,7)
(A,1,3)(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,7)(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)(A,1,15)
(A,1,7)(A,1,7)
(A,1,3)(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,7)(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)(A,1,7)
(A,1,3)(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,7)(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)
QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)(A,1,7)
(A,1,3)(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,7)(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)
QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)
Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)(A,1,7)
(A,1,3)(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,7)(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)
QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)
Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)(A,1,7)
(A,1,3)(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,7)(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,7)(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)
QuickSort(A, 1, 7)
QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,7)(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)
QuickSort(A, 1, 7)
QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)
Partition(A, 1, 7)
Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,7)(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)
QuickSort(A, 1, 7)
QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)
Partition(A, 1, 7)
Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,7)(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)
QuickSort(A, 1, 3)
QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)
QuickSort(A, 1, 3)
QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)
Partition(A, 1, 3)
Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)
QuickSort(A, 1, 3)
QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)
Partition(A, 1, 3)
Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)(A,1,1) (A,3,3)(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)
QuickSort(A, 1, 1)
QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)
QuickSort(A, 3, 3)
QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1) (A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)
QuickSort(A, 5, 7)
QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)
QuickSort(A, 5, 7)
QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)
Partition(A, 5, 7)
Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)
QuickSort(A, 5, 7)
QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)
Partition(A, 5, 7)
Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)(A,5,5) (A,7,7)(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5) (A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)
QuickSort(A, 5, 5)
QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5) (A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5) (A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)
QuickSort(A, 7, 7)
QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5) (A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)
QuickSort(A, 9, 15)
QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)
QuickSort(A, 9, 15)
QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)
Partition(A, 9, 15)
Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)
QuickSort(A, 9, 15)
QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)
Partition(A, 9, 15)
Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)
(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)
(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)
QuickSort(A, 9, 11)
QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)
(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)
(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)
QuickSort(A, 9, 11)
QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)
Partition(A, 9, 11)
Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)
(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)
(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)
QuickSort(A, 9, 11)
QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)
Partition(A, 9, 11)
Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)
(A,9,11)
(A,9,9)(A,9,9) (A,11,11)(A,11,11)
(A,13,15)
(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)
(A,9,11)
(A,9,9)
(A,9,9) (A,11,11)
(A,11,11)
(A,13,15)
(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)
QuickSort(A, 9, 9)
QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)
(A,9,11)
(A,9,9)
(A,9,9) (A,11,11)
(A,11,11)
(A,13,15)
(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)
(A,9,11)
(A,9,9)
(A,9,9) (A,11,11)
(A,11,11)
(A,13,15)
(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)
QuickSort(A, 11, 11)
QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)
(A,9,11)
(A,9,9)
(A,9,9) (A,11,11)
(A,11,11)
(A,13,15)
(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)
(A,9,11)
(A,9,9)
(A,9,9)
(A,11,11)
(A,11,11)
(A,13,15)
(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)
QuickSort(A, 13, 15)
QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)
(A,9,11)
(A,9,9)
(A,9,9)
(A,11,11)
(A,11,11)
(A,13,15)
(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)
QuickSort(A, 13, 15)
QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)
Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)
(A,9,11)
(A,9,9)
(A,9,9)
(A,11,11)
(A,11,11)
(A,13,15)
(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)
QuickSort(A, 13, 15)
QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)
Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)
(A,9,11)
(A,9,9)
(A,9,9)
(A,11,11)
(A,11,11)
(A,13,15)
(A,13,15)
(A,13,13)(A,13,13) (A,15,15)(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)
(A,9,11)
(A,9,9)
(A,9,9)
(A,11,11)
(A,11,11)
(A,13,15)
(A,13,15)
(A,13,13)
(A,13,13) (A,15,15)
(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)
QuickSort(A, 13, 13)
QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)
(A,9,11)
(A,9,9)
(A,9,9)
(A,11,11)
(A,11,11)
(A,13,15)
(A,13,15)
(A,13,13)
(A,13,13) (A,15,15)
(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)
(A,9,11)
(A,9,9)
(A,9,9)
(A,11,11)
(A,11,11)
(A,13,15)
(A,13,15)
(A,13,13)
(A,13,13) (A,15,15)
(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)
QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)
(A,9,11)
(A,9,9)
(A,9,9)
(A,11,11)
(A,11,11)
(A,13,15)
(A,13,15)
(A,13,13)
(A,13,13) (A,15,15)
(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Best-Case Behavior
QuickSort: Best-Case Behavior
QuickSort(A, p, r)
1 if p < r2 then q ← Partition(A, p, r)3 QuickSort(A, p, q − 1)4 QuickSort(A, q + 1, r)
QuickSort(A, 1, 15)QuickSort(A, 1, 7)QuickSort(A, 1, 3)QuickSort(A, 1, 1)QuickSort(A, 3, 3)QuickSort(A, 5, 7)QuickSort(A, 5, 5)QuickSort(A, 7, 7)QuickSort(A, 9, 15)QuickSort(A, 9, 11)QuickSort(A, 9, 9)QuickSort(A, 11, 11)QuickSort(A, 13, 15)QuickSort(A, 13, 13)QuickSort(A, 15, 15)
Partition(A, 1, 15)Partition(A, 1, 7)Partition(A, 1, 3)Partition(A, 5, 7)Partition(A, 9, 15)Partition(A, 9, 11)Partition(A, 13, 15)
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
5
3
11
4
2
5
12
6
7
7
6
8
4
9
9
10
13
11
14
12
15
13
10
14
8
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
6
3
2
4
7
5
5
6
4
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
3
1
1
2
2
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
7
5
5
6
6
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
13
11
14
12
15
13
10
14
12
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
11
9
9
10
10
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
15
13
13
14
14
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
(A,1,15)
(A,1,15)
(A,1,7)
(A,1,7)
(A,1,3)
(A,1,3)
(A,1,1)
(A,1,1)
(A,3,3)
(A,3,3)
(A,5,7)
(A,5,7)
(A,5,5)
(A,5,5)
(A,7,7)
(A,7,7)
(A,9,15)
(A,9,15)
(A,9,11)
(A,9,11)
(A,9,9)
(A,9,9)
(A,11,11)
(A,11,11)
(A,13,15)
(A,13,15)
(A,13,13)
(A,13,13)
(A,15,15)
(A,15,15)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 6 / 7
Randomized QuickSort
Randomized Quick Sort
RandomizedPartition(A, p, r)
1 i ← Random(p, r)2 exchange A[r ]↔ A[i ]3 return Partition(A, p, r)
RandomizedQuickSort(A, p, r)
1 if p < r2 then q ← RandomizedPartition(A, p, r)3 RandomizedQuickSort(A, p, q − 1)4 RandomizedQuickSort(A, q + 1, r)
Parosh Aziz Abdulla (Uppsala University) QuickSort September 12, 2012 7 / 7
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