Sorting Algorithms: Merge Sort and Quick Sort, Slides of Data Structures and Algorithms

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Merge Sort

Merging Two Sorted Arrays

All the work in merge sort is done at the merge step.

Example

1

13

24

26

2

15

27

38

Quick Sort

Quicksort idea

Choose a pivot.

Rearrange so thatpivot is in the “right”spot.

Recurse on each halfand conquer!

Quicksort algorithm

If array A has 1 (or 0) elements, then done.

Choose a

pivot element

x from A.

Divide A-{x} into two arrays:

B = {y

A | y≤x}

C = {y

A | y

x}

Result is B+{x}+C.

Recurse on arrays B and C

Place the pivot algorithm

Assume we have an array of size n

Pick a pivot x

Place the pivot x in the last place of the array

Have two pointers i = 0, j = n- while ( i <= j) {

while (A[j] > pivot) j--;while (A[i] < pivot) i--;swap (A[i], A[j]); } swap (A[i], pivot);

Example

Show how to place the pivot (choose middle element)in the right place

34 24 56 17 19 45 90 23 36

Naïve sorting algorithms

Bubble sort:

scan for flips, until all are fixed

Etc...

Naïve Sorting

for i=1 to n-

{ for j=0 to n-i-

if (A[j].compareTo(A[j+1])>0)

swap(A[j], A[j+1]);

if (no swaps) break; }

What happens if

All keys are equal?

Keys are sorted in reverse order?

Keys are sorted?

keys are randomly distributed?

Exercise: Count the number of operations in bubble sortand find a Big O analysis for bubble sort

Insertion sort^ •

Algorithm for

i =

1

to

n-

do

insert a[i] in the proper place

in

a[0:i-1]

Correctness

•Note: after i steps, the sub-array A[0:i] issorted

How fast is insertion sort?

of slides = O(#inversions)

very fast if array is nearly sorted to begin with

tmp

a[i];

for

(j = i;

j>0 && a[j-1]>tmp; j--)

a[j] = a[j-1];

a[j]

= tmp;

To insert a[i] into a[0:i-1],

slide

all elements larger

than a[i] to the right.

Sorting Comparison

Discuss the pros and cons of each of the naïve sortingalgorithms

Advanced Sorting