Sorting Algorithms Merge Sort Quick Sort Hairong Zhao New Jersey Institute of Technology.
Post on 18-Jan-2018
217 Views
Preview:
DESCRIPTION
Transcript
Sorting Algorithms
Merge Sort Quick Sort
Hairong Zhaohttp://web.njit.edu/~hz2
New Jersey Institute of Technology
2
Overview Divide and Conquer
Merge Sort
Quick Sort
3
Divide and Conquer1. Base Case, solve the problem directly
if it is small enough
2. Divide the problem into two or more similar and smaller subproblems
3. Recursively solve the subproblems
4. Combine solutions to the subproblems
4
Divide and Conquer - SortProblem: Input: A[left..right] – unsorted array of integers Output: A[left..right] – sorted in non-decreasing
order
5
Divide and Conquer - Sort1. Base case
at most one element (left ≥ right), return
2. Divide A into two subarrays: FirstPart, SecondPart Two Subproblems:
sort the FirstPart sort the SecondPart
3. Recursively sort FirstPart sort SecondPart
4. Combine sorted FirstPart and sorted SecondPart
6
Overview Divide and Conquer
Merge Sort
Quick Sort
7
Merge Sort: Idea
Merge
Recursively sort
Divide intotwo halves FirstPart SecondPart
FirstPart SecondPart
A
A is sorted!
8
Merge Sort: AlgorithmMerge-Sort (A, left, right)
if left ≥ right return
else
middle ← b(left+right)/2
Merge-Sort(A, left, middle)
Merge-Sort(A, middle+1, right)
Merge(A, left, middle, right)
Recursive Call
9A[middle]A[middle]A[left]A[left]
SortedSorted FirstPartFirstPart
Sorted Sorted SecondPartSecondPart
Merge-Sort: Merge
A[right]A[right]
mergemerge
A:A:
A:A:
SortedSorted
10
6 10 14 223 5 15 28L:L: R:R:
Temporary ArraysTemporary Arrays
5 15 28 30 6 10 145
Merge-Sort: Merge Example
2 3 7 8 1 4 5 6A:A:
11
Merge-Sort: Merge Example
3 5 15 28 30 6 10 14
L:L:
A:A:
3 15 28 30 6 10 14 22R:R:
i=0 j=0
k=0
2 3 7 8 1 4 5 6
1
12
Merge-Sort: Merge Example
1 5 15 28 30 6 10 14
L:L:
A:A:
3 5 15 28 6 10 14 22R:R:
k=1
2 3 7 8 1 4 5 6
2
i=0 j=1
13
Merge-Sort: Merge Example
1 2 15 28 30 6 10 14
L:L:
A:A:
6 10 14 22R:R:
i=1
k=2
2 3 7 8 1 4 5 6
3
j=1
14
Merge-Sort: Merge Example
1 2 3 6 10 14
L:L:
A:A:
6 10 14 22R:R:
i=2 j=1
k=3
2 3 7 8 1 4 5 6
4
15
Merge-Sort: Merge Example
1 2 3 4 6 10 14
L:L:
A:A:
6 10 14 22R:R:
j=2
k=4
2 3 7 8 1 4 5 6
i=2
5
16
Merge-Sort: Merge Example
1 2 3 4 5 6 10 14
L:L:
A:A:
6 10 14 22R:R:
i=2 j=3
k=5
2 3 7 8 1 4 5 6
6
17
Merge-Sort: Merge Example
1 2 3 4 5 6 14
L:L:
A:A:
6 10 14 22R:R:
k=6
2 3 7 8 1 4 5 6
7
i=2 j=4
18
Merge-Sort: Merge Example
1 2 3 4 5 6 7 14
L:L:
A:A:
3 5 15 28 6 10 14 22R:R:
2 3 7 8 1 4 5 6
8
i=3 j=4
k=7
19
Merge-Sort: Merge Example
1 2 3 4 5 6 7 8
L:L:
A:A:
3 5 15 28 6 10 14 22R:R:
2 3 7 8 1 4 5 6
i=4 j=4
k=8
20
Merge(A, left, middle, right)1. n1 ← middle – left + 12. n2 ← right – middle3. create array L[n1], R[n2]4. for i ← 0 to n1-1 do L[i] ← A[left +i]5. for j ← 0 to n2-1 do R[j] ← A[middle+j]6. k ← i ← j ← 07. while i < n1 & j < n2 8. if L[i] < R[j] 9. A[k++] ← L[i++]10. else11. A[k++] ← R[j++]12. while i < n1
13. A[k++] ← L[i++]14. while j < n2
15. A[k++] ← R[j++] n = n1+n2
Space: n
Time : cn for some constant c
21
6 2 8 4 3 7 5 16 2 8 4 3 7 5 1
Merge-Sort(A, 0, 7)Divide
A:
22
6 2 8 4
3 7 5 1
6 2 8 4
Merge-Sort(A, 0, 3) , divideA:
Merge-Sort(A, 0, 7)
23
3 7 5 1
8 4
6 26 2
Merge-Sort(A, 0, 1), divideA:
Merge-Sort(A, 0, 7)
24
3 7 5 1
8 4
6
2
Merge-Sort(A, 0, 0) , base caseA:
Merge-Sort(A, 0, 7)
25
3 7 5 1
8 4
6 2
Merge-Sort(A, 0, 0), returnA:
Merge-Sort(A, 0, 7)
26
3 7 5 1
8 4
6
2
Merge-Sort(A, 1, 1), base caseA:
Merge-Sort(A, 0, 7)
27
3 7 5 1
8 4
6 2
Merge-Sort(A, 1, 1), returnA:
Merge-Sort(A, 0, 7)
28
3 7 5 1
8 4
2 6
Merge(A, 0, 0, 1)A:
Merge-Sort(A, 0, 7)
29
3 7 5 1
8 42 6
Merge-Sort(A, 0, 1), returnA:
Merge-Sort(A, 0, 7)
30
3 7 5 1
8 4
2 6
Merge-Sort(A, 2, 3)
48
, divideA:
Merge-Sort(A, 0, 7)
31
3 7 5 1
4
2 6
8
Merge-Sort(A, 2, 2), base caseA:
Merge-Sort(A, 0, 7)
32
3 7 5 1
4
2 6
8
Merge-Sort(A, 2, 2), returnA:
Merge-Sort(A, 0, 7)
33
4
2 6
8
Merge-Sort(A, 3, 3), base caseA:
Merge-Sort(A, 0, 7)
34
3 7 5 1
4
2 6
8
Merge-Sort(A, 3, 3), returnA:
Merge-Sort(A, 0, 7)
35
3 7 5 1
2 6
4 8
Merge(A, 2, 2, 3)A:
Merge-Sort(A, 0, 7)
36
3 7 5 1
2 6 4 8
Merge-Sort(A, 2, 3), returnA:
Merge-Sort(A, 0, 7)
37
3 7 5 1
2 4 6 8
Merge(A, 0, 1, 3)A:
Merge-Sort(A, 0, 7)
38
3 7 5 12 4 6 8Merge-Sort(A, 0, 3), return
A:
Merge-Sort(A, 0, 7)
39
3 7 5 1
2 4 6 8Merge-Sort(A, 4, 7)
A:
Merge-Sort(A, 0, 7)
40
1 3 5 7
2 4 6 8A:Merge (A, 4, 5, 7)
Merge-Sort(A, 0, 7)
41
1 3 5 72 4 6 8Merge-Sort(A, 4, 7), return
A:
Merge-Sort(A, 0, 7)
42
1 2 3 4 5 6 7 8Merge(A, 0, 3, 7)
A:
Merge-Sort(A, 0, 7)Merge-Sort(A, 0, 7), done!
43
Merge-Sort Analysis cn
2 × cn/2 = cn
4 × cn/4 = cn
n/2 × 2c = cn
log n levels
• Total running time: (nlogn)• Total Space: (n)
Total: cn log n
n
n/2 n/2
n/4 n/4 n/4 n/4
2 2 2
44
Merge-Sort SummaryApproach: divide and conquerTime
Most of the work is in the merging Total time: (n log n)
Space: (n), more space than other sorts.
45
Overview Divide and Conquer
Merge Sort
Quick Sort
46
Quick Sort Divide:
Pick any element p as the pivot, e.g, the first element
Partition the remaining elements into FirstPart, which contains all elements < pSecondPart, which contains all elements ≥ p
Recursively sort the FirstPart and SecondPart
Combine: no work is necessary since sorting is done in place
47
Quick Sort
x < p p p ≤ x
PartitionFirstPart SecondPart
ppivot
A:
Recursive call
x < p p p ≤ x
SortedFirstPart
SortedSecondPart
SortedSorted
48
Quick SortQuick-Sort(A, left, right)if left ≥ right return
else middle ← Partition(A, left,
right) Quick-Sort(A, left, middle–1 ) Quick-Sort(A, middle+1, right)
end if
49
Partition
p
p x < p p ≤ x
p p ≤ xx < p
A:A:
A:A:
A:A:p
50
Partition Example
A:A: 4 8 6 3 5 1 7 2
51
Partition Example
A:A: 4 8 6 3 5 1 7 2
i=0i=0
j=1j=1
52
Partition Example
A:A:
j=1j=1
4 8 6 3 5 1 7 2
i=0i=0
8
53
Partition Example
A:A: 4 8 6 3 5 1 7 26
i=0i=0
j=2j=2
54
Partition Example
A:A: 4 8 6 3 5 1 7 2
i=0i=0
383
j=3j=3
i=1i=1
55
Partition Example
A:A: 4 3 6 8 5 1 7 2
i=1i=1
5
j=4j=4
56
Partition Example
A:A: 4 3 6 8 5 1 7 2
i=1i=1
1
j=5j=5
57
Partition Example
A:A: 4 3 6 8 5 1 7 2
i=2i=2
1 6
j=5j=5
58
Partition Example
A:A: 4 3 8 5 7 2
i=2i=2
1 6 7
j=6j=6
59
Partition Example
A:A: 4 3 8 5 7 2
i=2i=2
1 6 22 8
i=3i=3
j=7j=7
60
Partition Example
A:A: 4 3 2 6 7 8
i=3i=3
1 5
j=8j=8
61
Partition Example
A:A: 4 1 6 7 8
i=3i=3
2 542 3
62
A:A: 3 6 7 81 542
x < 4x < 4 4 ≤ x4 ≤ x
pivot incorrect position
Partition Example
63
Partition(A, left, right)1. x ← A[left]2. i ← left3. for j ← left+1 to right4. if A[j] < x then 5. i ← i + 16. swap(A[i], A[j])7. end if8. end for j9. swap(A[i], A[left])10. return i
n = right – left +1 Time: cn for some constant c Space: constant
64
4 8 6 3 5 1 7 22 3 1 5 6 7 84
Quick-Sort(A, 0, 7)Partition
A:
65
2 3 1
5 6 7 84
2 1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 0, 2)
A:
, partition
66
2
5 6 7 84
1
1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 0, 0) , base case, return
67
2
5 6 7 84
1
33
Quick-Sort(A, 0, 7)Quick-Sort(A, 1, 1) , base case
68
5 6 7 842 1 3
2 1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 2, 2), returnQuick-Sort(A, 0, 2), return
69
42 1 3
5 6 7 86 7 85
Quick-Sort(A, 0, 7)Quick-Sort(A, 2, 2), returnQuick-Sort(A, 4, 7) , partition
70
4
5
6 7 87 866
2 1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 5, 7) , partition
71
4
5
6
7 887
2 1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 6, 7) , partition
72
4
5
6
7
2 1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 7, 7)
8
, return, base case
8
73
4
5
6 87
2 1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 6, 7) , return
74
4
5
2 1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 5, 7) , return
6 87
75
42 1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 4, 7) , return
5 6 87
76
42 1 3
Quick-Sort(A, 0, 7)Quick-Sort(A, 0, 7) , done!
5 6 87
77
Quick-Sort: Best Case Even Partition
Total time: (nlogn)
cn
2 × cn/2 = cn
4 × c/4 = cn
n/3 × 3c = cn
log n levels
n
n/2 n/2
n/4
3 3 3
n/4n/4n/4
78
cn
c(n-1)
3c
2c
n
n-1
n-2
3
2
c(n-2)
Happens only if input is sortd input is reversely sorted
Quick-Sort: Worst Case Unbalanced Partition
Total time: (n2)
79
Quick-Sort: an Average Case Suppose the split is 1/10 : 9/10
Quick-Sort: an Average Case
cn
cn
cn
≤cn
n
0.1n 0.9n
0.01n 0.09n 0.09n
Total time: (nlogn)
0.81n
2
2
log10n
log10/9n
≤cn
80
Quick-Sort Summary Time
Most of the work done in partitioning. Average case takes (n log(n)) time. Worst case takes (n2) time
Space Sorts in-place, i.e., does not require additional
space
81
Summary Divide and Conquer
Merge-Sort Most of the work done in Merging (n log(n)) time (n) space
Quick-Sort Most of the work done in partitioning Average case takes (n log(n)) time Worst case takes (n2) time (1) space
82
Homework1. What is the running time of Merge-Sort if the
array is already sorted? What is the best case running time of Merge-Sort?
2. Demonstrate the working of Partition on sequence (13, 5, 14, 11, 16, 12, 1, 15). What is the value of i returned at the completion of Partition?
top related