data structure notes for computer engineering note no:2

7/31/2019 data structure notes for computer engineering note no:2

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Sorting

Sorting is the process of arranging a set of

elements in a specific order.

Sorting can be classified in to

Internal sorting

External sorting


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Internal

All of the data are

held in the primarymemory during the

sorting process.

External

Uses primary

memory for the datacurrently being

sorted and

secondary storage

for any data thatdoes not fit in

primary memory.

Classification


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Bubble Sort

Traverse a collection of elements

Move from the front to the end

Bubble the largest value to the end using pair-

wise comparisons and swapping

512354277 101

1 2 3 4 5 6


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512354277 101

1 2 3 4 5 6

Swap

42 77

512357742 101

1 2 3 4 5 6

35 77

512773542 101

1 2 3 4 5 6

Swap

12 77

577123542 101

1 2 3 4 5 6

No need to swap


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577123542 101

1 2 3 4 5 6

Swap

5 101

77123542 5

1 2 3 4 5 6

101

Largest value correctly placed


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Notice that only the largest value is correctly

placed

All other values are still out of order

So we need to repeat this process

77123542 5

1 2 3 4 5 6

101

Largest value correctly placed


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The Algorithm

index A[index + 1]) then

Swap(A[index], A[index + 1])

end ifindex


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Insertion sort eg:


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The Algorithm

for i=2 to n do

begin

j := i

while (a[j] < a[j-1])

begin

swap ( a[j],a[j-1] )

j = j +1

end

endComplexity of insertion sort:

In best case= 0(n);worst case=average case=0(n2)


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Selection Sort

The smallest element is selected from the

unsorted sublist and exchange with the

element at thebeginning of the unsorted list.


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Algorithm

set current to 0

loop (until last element sorted)

set smallest to current

set index to current + 1

loop (index < last)

if (index key < smallest key)set smallest to index

increment index

end loop

smallest selected; exchange with current element.

exchange (current, smallest)increment current

end loop

End selection Sort


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Selection Sort

Original list23 78 45 8 32 56

Unsorted

After pass 18 78 45 23 32 56

UnsortedSorted

After pass 28 23 45 78 32 56

UnsortedSorted

After pass 38 23 32 78 45 56

UnsortedSorted

After pass 48 23 32 45 78 56

UnsortedSorted

After pass 58 23 32 45 56 78

Sorted


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Quicksort

To sort a[left...right]:if left < right:

Partition a[left...right] such that:all a[left...p-1] are less than a[p], and

all a[p+1...right] are >= a[p]

Quicksort a[left...p-1]

Quicksort a[p+1...right]

Terminate


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A key step in the Quicksort algorithm ispartitioning the array

We choose some (any) numberpin the arrayto use as a pivot

We partition the array into three parts:

p

numbers

less thanp

numbers greater

than or equal top

p


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Choose an array value (say, the first) to use asthe pivot

Starting from the left end, find the firstelement that is greater than or equal to thepivot

Searching backward from the right end, findthe first element that is less than the pivot

Interchange (swap) these two elements Repeat, searching from where we left off, until

done

Quicksort is usuallyO(n log2n)


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To partition a[left...right]:

Set p = a[left], l = left + 1, r = right;

while l < r, do

while l < right & a[l] < p, set l = l + 1while r > left & a[r] >= p, set r = r - 1if l < r, swap a[l] and a[r]

Set a[left] = a[r], a[r] = p

Terminate


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Mergesort

A divide-and-conquer algorithm: Divide the unsorted array into 2 halves until

the sub-arrays only contain one element

Merge the sub-problem solutions together:

Compare the sub-arrays first elements

Remove the smallest element and put it into theresult array

Continue the process until all elements have beenput into the result array

37 23 6 89 15 12 2 19


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2398


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2398

Merge


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2398

23

Merge


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2398

23 98

Merge


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23 98


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2398 45 14

Merge

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14

Merge

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45

Merge

23 98 14


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2398 45 14

Merge

98 451423


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Merge

98 14

14

23 45


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Merge

23 14

14 23

98 45


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674523 14 6 3398 42

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2398 45 14

Merge

23 98 4514

14 23 45


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674523 14 6 3398 42

4523 1498

2398 45 14

Merge

23 98 4514

14 23 45 98


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674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

23 98 4514

14 23 45 98


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674523 14 6 3398 42

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2398 45 14

676 33 42

676

23 98 4514

14 23 45 98


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674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676

Merge

23 98 4514

14 23 45 98


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674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676

6

Merge

23 98 4514

14 23 45 98


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674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676

67

Merge

23 98 4514 6

14 23 45 98


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674523 14 6 3398 42

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2398 45 14

676 33 42

676 33 42

23 98 4514 676

14 23 45 98


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674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676 33 42

Merge

23 98 4514 676

14 23 45 98


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674523 14 6 3398 42

674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676 33 42

Merge

3323 98 4514 676

14 23 45 98


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674523 14 6 3398 42

674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676 33 42

Merge

4223 98 4514 676 33

14 23 45 98


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674523 14 6 3398 42

674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676 33 42

Merge

23 98 4514 676 4233

14 23 45 98


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674523 14 6 3398 42

674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676 33 42

Merge

23 98 4514 6 4233

14 23 45 98 6

67


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674523 14 6 3398 42

674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676 33 42

Merge

23 98 4514 6 33

14 23 45 98 6 33

67 42


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674523 14 6 3398 42

674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676 33 42

Merge

23 98 4514 6 4233

14 23 45 98 6 33 42

67


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674523 14 6 3398 42

674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676 33 42

Merge

23 98 4514 676 4233

14 23 45 98 6 33 42 67


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674523 14 6 3398 42

674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676 33 42

Merge

23 98 4514 676 4233

23 45 98 33 42 6714 6


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674523 14 6 3398 42

674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676 33 42

Merge

23 98 4514 676 4233

23 45 98 6 42 67

6

14 33


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674523 14 6 3398 42

674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676 33 42

Merge

23 98 4514 676 4233

14 45 98 6 42 67

6 14

23 33


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674523 14 6 3398 42

674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676 33 42

Merge

23 98 4514 676 4233

14 23 98 6 42 67

6 14 23

45 33


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674523 14 6 3398 42

674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676 33 42

Merge

23 98 4514 676 4233

14 23 98 6 33 67

6 14 23 33

45 42


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674523 14 6 3398 42

674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676 33 42

Merge

23 98 4514 676 4233

14 23 98 6 33 42

6 14 23 33 42

45 67


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674523 14 6 3398 42

674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676 33 42

Merge

23 98 4514 676 4233

14 23 45 6 33 42

6 14 23 33 42 45

98 67


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674523 14 6 3398 42

674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676 33 42

Merge

23 98 4514 676 4233

14 23 45 98 6 33 42 67

6 14 23 33 42 45 67


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674523 14 6 3398 42

674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676 33 42

Merge

23 98 4514 676 4233

14 23 45 98 6 33 42 67

6 14 23 33 42 45 67 98


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674523 14 6 3398 42

674523 14 6 3398 42

4523 1498

2398 45 14

676 33 42

676 33 42

23 98 4514 676 4233

14 23 45 98 6 33 42 67

6 14 23 33 42 45 67 98


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674523 14 6 3398 42

6 14 23 33 42 45 67 98


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Heap Sort

It is an improved version of selection sort in

which the largest element is selected and

exchanged with the last element in the

unsorted list.


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set index to 1loop (heap built)

reheapUp (heap, index)

increment index

end loop

heap created. Now sort it.

set sorted to last

loop (until all data sorted)

exchange (heap, 0, sorted)

decrement sorted

reheapDown (heap, 0, sorted)

end loop

End heapSort

Original list 23 78 45 8 32 56


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Unsorted

After heap78 32 56 8 23 45

HeapAfter pass 1and reheap

56 32 45 8 23 78

SortedHeapAfter pass 2and reheap

45 32 23 8 56 78


32 8 23 45 56 78

SortedHeap

After pass 4and reheap

23 8 32 45 56 78


8 23 32 45 56 78

SortedHeap

After pass 6and reheap

8 23 32 45 56 78

Sorted


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Heap Sort: Summary

Heap sort uses a heap data structure to

improve selection sort and make the running

time asymptotically optimal

Running time is O(n log n) like merge sort,

but unlike selection, insertion, or bubble sorts

Sorts in place like insertion, selection or

bubble sorts, but unlike merge sort

data structure notes for computer engineering note no:2

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