![Merge Sort
Sort A[0] to A[n/2-1]
Sort A[n/2] to A[n-1]
Merge sorted halves into B[0..n-1]
How do we sort the halves?
Recursively, using the same strategy!](https://image.slidesharecdn.com/nptel-week2-module5-mergesort-230920041709-91e75b61/85/merge-sort-12-320.jpg)
![Merging
function Merge(A,m,B,n,C)
// Merge A[0..m-1], B[0..n-1] into C[0..m+n-1]
i = 0; j = 0; k = 0;
// Current positions in A,B,C respectively
while (k < m+n)
// Case 0: One of the two lists is empty
if (i==m) {j++; k++;}
if (j==n) {i++; k++;}
// Case 1: Move head of A into C
if (A[i] <= B[j]) { C[k] = B[j]; j++; k++;}
// Case 2: Move head of B into C
if (A[i] > B[j]) {C[k] = B[j]; j++; k++;}](https://image.slidesharecdn.com/nptel-week2-module5-mergesort-230920041709-91e75b61/85/merge-sort-31-320.jpg)
![Merge Sort
To sort A[0..n-1] into B[0..n-1]
If n is 1, nothing to be done
Otherwise
Sort A[0..n/2-1] into L (left)
Sort A[n/2..n-1] into R (right)
Merge L and R into B](https://image.slidesharecdn.com/nptel-week2-module5-mergesort-230920041709-91e75b61/85/merge-sort-32-320.jpg)
![Merge Sort
function MergeSort(A,left,right,B)
// Sort the segment A[left..right-1] into B
if (right - left == 1) // Base case
B[0] = A[left]
if (right - left > 1) // Recursive call
mid = (left+right)/2
MergeSort(A,left,mid,L)
MergeSort(A,mid,right,R)
Merge(L,mid-left,R,right-mid,B)](https://image.slidesharecdn.com/nptel-week2-module5-mergesort-230920041709-91e75b61/85/merge-sort-33-320.jpg)
merge sort
- 1. DESIGN AND ANALYSIS OF ALGORITHMS Merge Sort MADHAVAN MUKUND, CHENNAI MATHEMATICAL INSTITUTE http://www.cmi.ac.in/~madhavan NPTEL MOOC,JAN-FEB 2015 Week 2, Module 5
- 2. O(n2) sorting algorithms Selection sort and insertion sort are both O(n2) O(n2) sorting is infeasible for n over 100000
- 3. A different strategy? Divide array in two equal parts Separately sort left and right half Combine the two sorted halves to get the full array sorted
- 4. Combining sorted lists Given two sorted lists A and B, combine into a sorted list C Compare first element of A and B Move it into C Repeat until all elements in A and B are over Merging A and B
- 5. Merging two sorted lists 32 74 89 21 55 64
- 6. Merging two sorted lists 32 74 89 21 55 64 21
- 7. Merging two sorted lists 32 74 89 21 55 64 21 32
- 8. Merging two sorted lists 32 74 89 21 55 64 21 32 55
- 9. Merging two sorted lists 32 74 89 21 55 64 21 32 55 64
- 10. Merging two sorted lists 32 74 89 21 55 64 21 32 55 64 74
- 11. Merging two sorted lists 32 74 89 21 55 64 21 32 55 64 74 89
- 12. Merge Sort Sort A[0] to A[n/2-1] Sort A[n/2] to A[n-1] Merge sorted halves into B[0..n-1] How do we sort the halves? Recursively, using the same strategy!
- 13. Merge Sort 43 32 22 78 63 57 91 13
- 14. Merge Sort 43 32 22 78 63 57 91 13 43 32 22 78
- 15. Merge Sort 43 32 22 78 63 57 91 13 43 32 22 78 63 57 91 13
- 16. Merge Sort 43 32 22 78 63 57 91 13 43 32 22 78 63 57 91 13 43 32 22 78
- 17. Merge Sort 43 32 22 78 63 57 91 13 43 32 22 78 63 57 91 13 43 32 22 78 63 57 91 13
- 18. Merge Sort 43 32 22 78 63 57 91 13 43 32 22 78 63 57 91 13 43 32 22 78 63 57 91 13 43 32
- 19. Merge Sort 43 32 22 78 63 57 91 13 43 32 22 78 63 57 91 13 43 32 22 78 63 57 91 13 43 32 22 78
- 20. Merge Sort 43 32 22 78 63 57 91 13 43 32 22 78 63 57 91 13 43 32 22 78 63 57 91 13 43 32 22 78 63 57
- 21. Merge Sort 43 32 22 78 63 57 91 13 43 32 22 78 63 57 91 13 43 32 22 78 63 57 91 13 43 32 22 78 63 57 91 13
- 22. Merge Sort 43 32 22 78 63 57 91 13 43 32 22 78 63 57 91 13 22 78 63 57 91 13 43 32 22 78 63 57 91 13 32 43
- 23. Merge Sort 43 32 22 78 63 57 91 13 43 32 22 78 63 57 91 13 63 57 91 13 43 32 22 78 63 57 91 13 32 43 22 78
- 24. Merge Sort 43 32 22 78 63 57 91 13 43 32 22 78 63 57 91 13 91 13 43 32 22 78 63 57 91 13 32 43 22 78 57 63
- 25. Merge Sort 43 32 22 78 63 57 91 13 43 32 22 78 63 57 91 13 43 32 22 78 63 57 91 13 32 43 22 78 57 63 13 91
- 26. Merge Sort 43 32 22 78 63 57 91 13 63 57 91 13 43 32 22 78 63 57 91 13 32 43 22 78 57 63 13 91 22 32 43 78
- 27. Merge Sort 43 32 22 78 63 57 91 13 43 32 22 78 63 57 91 13 32 43 22 78 57 63 13 91 22 32 43 78 13 57 63 91
- 28. Merge Sort 43 32 22 78 63 57 91 13 32 43 22 78 57 63 13 91 22 32 43 78 13 57 63 91 13 22 32 43 57 63 78 91
- 29. Divide and conquer Break up problem into disjoint parts Solve each part separately Combine the solutions efficiently
- 30. Merging sorted lists Combine two sorted lists A and B into C If A is empty, copy B into C If B is empty, copy A into C Otherwise, compare first element of A and B and move the smaller of the two into C Repeat until all elements in A and B have been moved
- 31. Merging function Merge(A,m,B,n,C) // Merge A[0..m-1], B[0..n-1] into C[0..m+n-1] i = 0; j = 0; k = 0; // Current positions in A,B,C respectively while (k < m+n) // Case 0: One of the two lists is empty if (i==m) {j++; k++;} if (j==n) {i++; k++;} // Case 1: Move head of A into C if (A[i] <= B[j]) { C[k] = B[j]; j++; k++;} // Case 2: Move head of B into C if (A[i] > B[j]) {C[k] = B[j]; j++; k++;}
- 32. Merge Sort To sort A[0..n-1] into B[0..n-1] If n is 1, nothing to be done Otherwise Sort A[0..n/2-1] into L (left) Sort A[n/2..n-1] into R (right) Merge L and R into B
- 33. Merge Sort function MergeSort(A,left,right,B) // Sort the segment A[left..right-1] into B if (right - left == 1) // Base case B[0] = A[left] if (right - left > 1) // Recursive call mid = (left+right)/2 MergeSort(A,left,mid,L) MergeSort(A,mid,right,R) Merge(L,mid-left,R,right-mid,B)