• By:-

• Gagan.R.Popale-2BV13CS034

• Ankit Gaonkar -2BV13CS014

• Anung Tajo-2BV13CS015

• Alan S Malekar-2BV13CS010

Sorting

WHAT IS SORTING?

Sorting is the process of putting a list or a group of items in a specific order . Some common sorting criteria are: alphabetical or numerical.

Ex:- Merge sort, Quick sort etc

Merge sort and Quick sort use : Divide And Conqure Technique

3

Divide the problem into a number of sub-problems

◦ Similar sub-problems of smaller size

Conquer the sub-problems

◦ Solve the sub-problems recursively

◦ Sub-problem size small enough solve the problems in

straightforward manner

Combine the solutions of the sub-problems

◦ Obtain the solution for the original problem

Merge Sort

In the Beginning…

Invented by

John von Neumann (1903-1957)

Follows divide and conquer paradigm.

Developed merge sort for EDVAC in 1945

Merging

The key to Merge Sort is merging two sorted lists into one, such that if you have two lists X (x1x2

…xm) and Y(y1y2

…yn) the resulting list is Z(z1z2

…zm+n)

Example:

L1 = { 3 8 9 } L2 = { 1 5 7 }

merge(L1, L2) = { 1 3 5 7 8 9 }

Divide And Conquer

1.Divide: Divide the unsorted list into two sub lists of about half the size.

2.Conquer: Sort each of the two sub lists recursively until we have list sizes of length 1,in which case the list itself is returned.

3.Combine: Merge the two-sorted sub lists back into one sorted list.

Merge sort algorithm

Merge-Sort (A, n)

if n=1 return

else

n1 ← n2 ← n/2

create array L[n1], R[n2]

for i ← 0 to n1-1 do L[i] ← A[i]

for j ← 0 to n2-1 do R[j] ← A[n1+j]

Merge-Sort(L, n1)

Merge-Sort(R, n2)

Merge(A, L, n1, R, n2 )

Merge Sort Example

99 6 86 15 58 35 86 4 0

99 6 86 15 58 35 86 4 0

Merge Sort Example

99 6 86 15 58 35 86 4 0

99 6 86 15 58 35 86 4 0

86 1599 6 58 35 86 4 0

Merge Sort Example

99 6 86 15 58 35 86 4 0

99 6 86 15 58 35 86 4 0

86 1599 6 58 35 86 4 0

99 6 86 15 58 35 86 4 0

Merge Sort Example

99 6 86 15 58 35 86 4 0

99 6 86 15 58 35 86 4 0

86 1599 6 58 35 86 4 0

99 6 86 15 58 35 86 4 0

4 0

Merge Sort Example

99 6 86 15 58 35 86 4 0

Merge Sort Example

99 6 86 15 58 35 86 0 4

4 0Merge

Merge Sort Example

15 866 99 58 35 0 4 86

99 6 86 15 58 35 86 0 4

Merge

Merge Sort Example

6 15 86 99 0 4 35 58 86

15 866 99 58 35 0 4 86

Merge

Merge Sort Example

0 4 6 15 35 58 86 86 99

6 15 86 99 0 4 35 58 86

Merge

Merge Sort Example

0 4 6 15 35 58 86 86 99

Implementing Merge Sort

There are two basic ways to implement merge sort:

In Place: Merging is done with only the input array

Double Storage: Merging is done with a temporary array of the same size as the input array.

20

Merge-Sort Analysis

• Time, merging

log n levels

• Total running time: order of nlogn• Total Space: order of n

Total time for merging: cn log n

n

n/2 n/2

n/4 n/4 n/4 n/4

Additional

Merge sort’s merge operation is useful in online sorting, where the list to be sorted is received a piece at a time,instead of all at the beginning..

In this We sort each new piece that is received using any sorting algorithm, and then merge it into our sorted list so far using the merge operation.

Finally

Best Case, Average Case, and Worst Case = O(N logN)

• Storage Requirement:

Double that needed to hold the array to be sorted.