Data Structure and Algorithms Merge Sort

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Merge sort is a divide and conquer algorithm that works as follows: 1) Divide the array to be sorted into two halves recursively until single element subarrays are reached. 2) Merge the subarrays in a way that combines them in a sorted order. 3) The merging process involves taking the first element of each subarray and comparing them to place the smaller element in the final array until all elements are merged. Merge sort runs in O(n log n) time in all cases making it one of the most efficient sorting algorithms.

Merge Sort
 Divide and Conquer
 Recursive in structure
 Divide the problem into sub-problems that are similar
to the original but smaller in size
 Conquer the sub-problems by solving them recursively.
If they are small enough, just solve them in a
straightforward manner.
 Combine the solutions to create a solution to the
original problem

An Example: Merge Sort
Sorting Problem: Sort a sequence of n elements into
non-decreasing order.
 Divide: Divide the n-element sequence to be
sorted into two subsequences of n/2 elements each
 Conquer: Sort the two subsequences recursively
using merge sort.
 Combine: Merge the two sorted subsequences to
produce the sorted answer.

Merge Sort – Example
18 26 32 6 43 15 9 1
18 26 32 6 43 15 9 1
18 26 32 6 43 15 9 1
2618 632 1543 19
18 26 32 6 43 15 9 1
18 26 32 6 43 15 9 1
18 26 326 15 43 1 9
6 18 26 32 1 9 15 43
1 6 9 15 18 26 32 43
18 26
18 26
18 26
32
32
6
6
32 6
18 26 32 6
43
43
15
15
43 15
9
9
1
1
9 1
43 15 9 1
18 26 32 6 43 15 9 1
18 26 632
626 3218
1543 19
1 915 43
16 9 1518 26 32 43
Original Sequence Sorted Sequence

Most read

j
Merge – Example
6 8 26 32 1 9 42 43… …A
k
6 8 26 32 1 9 42 43
k k k k k k k
i i i i
 
i j j j j
6 8 26 32 1 9 42 43
1 6 8 9 26 32 42 43
k
L R

Analysis of Merge Sort
 Running time T(n) of Merge Sort:
 Divide: computing the middle takes (1)
 Conquer: solving 2 sub-problems takes 2T(n/2)
 Combine: merging n elements takes (n)
 Total:
T(n) = (1) if n = 1
T(n) = 2T(n/2) + (n) if n > 1
T(n) = 2 T(n/2) + n
= 2 ((n/2)log(n/2) + (n/2)) + n
= n (log(n/2)) + 2n
= n log n – n + 2n
= n log n + n
= O(n log n )

Comparing the Algorithms
Best Average Worst
Case Case Case
Bubble Sort O(n) O(n2) O(n2)
Insertion Sort O(n) O(n2) O(n2)
Selection Sort O(n2) O(n2) O(n2)
Merge Sort O(n log n) O(n log n) O(n log n)
Quick Sort O(n log n) O(n log n) O(n2)
Heap Sort O(n log n) O(n log n) O(n log n)

Merge-Sort (A, p, r)
INPUT: a sequence of n numbers stored in array A
OUTPUT: an ordered sequence of n numbers
MergeSort (A, p, r) // sort A[p..r] by divide & conquer
1 if p < r
2 then q  (p+r)/2
3 MergeSort (A, p, q)
4 MergeSort (A, q+1, r)
5 Merge (A, p, q, r) // merges A[p..q] with A[q+1..r]
Initial Call: MergeSort(A, 1, n)

Procedure Merge
Merge(A, p, q, r)
1 n1  q – p + 1
2 n2  r – q
3 for i  1 to n1
4 do L[i]  A[p + i – 1]
5 for j  1 to n2
6 do R[j]  A[q + j]
7 L[n1+1]  
8 R[n2+1]  
9 i  1
10 j  1
11 for k p to r
12 do if L[i]  R[j]
13 then A[k]  L[i]
14 i  i + 1
15 else A[k]  R[j]
16 j  j + 1
Input: Array containing sorted subarrays
A[p..q] and A[q+1..r].
Output: Merged sorted subarray in A[p..r].

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Data Structure and Algorithms Merge Sort

1.  Merge Sort

2. Merge Sort  Divide and Conquer  Recursive in structure  Divide the problem into sub-problems that are similar to the original but smaller in size  Conquer the sub-problems by solving them recursively. If they are small enough, just solve them in a straightforward manner.  Combine the solutions to create a solution to the original problem

3. An Example: Merge Sort Sorting Problem: Sort a sequence of n elements into non-decreasing order.  Divide: Divide the n-element sequence to be sorted into two subsequences of n/2 elements each  Conquer: Sort the two subsequences recursively using merge sort.  Combine: Merge the two sorted subsequences to produce the sorted answer.

4. Merge Sort – Example 18 26 32 6 43 15 9 1 18 26 32 6 43 15 9 1 18 26 32 6 43 15 9 1 2618 632 1543 19 18 26 32 6 43 15 9 1 18 26 32 6 43 15 9 1 18 26 326 15 43 1 9 6 18 26 32 1 9 15 43 1 6 9 15 18 26 32 43 18 26 18 26 18 26 32 32 6 6 32 6 18 26 32 6 43 43 15 15 43 15 9 9 1 1 9 1 43 15 9 1 18 26 32 6 43 15 9 1 18 26 632 626 3218 1543 19 1 915 43 16 9 1518 26 32 43 Original Sequence Sorted Sequence

5. Merge-Sort (A, p, r) INPUT: a sequence of n numbers stored in array A OUTPUT: an ordered sequence of n numbers MergeSort (A, p, r) // sort A[p..r] by divide & conquer 1 if p < r 2 then q  (p+r)/2 3 MergeSort (A, p, q) 4 MergeSort (A, q+1, r) 5 Merge (A, p, q, r) // merges A[p..q] with A[q+1..r] Initial Call: MergeSort(A, 1, n)

6. Procedure Merge Merge(A, p, q, r) 1 n1  q – p + 1 2 n2  r – q 3 for i  1 to n1 4 do L[i]  A[p + i – 1] 5 for j  1 to n2 6 do R[j]  A[q + j] 7 L[n1+1]   8 R[n2+1]   9 i  1 10 j  1 11 for k p to r 12 do if L[i]  R[j] 13 then A[k]  L[i] 14 i  i + 1 15 else A[k]  R[j] 16 j  j + 1 Input: Array containing sorted subarrays A[p..q] and A[q+1..r]. Output: Merged sorted subarray in A[p..r].

7. j Merge – Example 6 8 26 32 1 9 42 43… …A k 6 8 26 32 1 9 42 43 k k k k k k k i i i i   i j j j j 6 8 26 32 1 9 42 43 1 6 8 9 26 32 42 43 k L R

8. Analysis of Merge Sort  Running time T(n) of Merge Sort:  Divide: computing the middle takes (1)  Conquer: solving 2 sub-problems takes 2T(n/2)  Combine: merging n elements takes (n)  Total: T(n) = (1) if n = 1 T(n) = 2T(n/2) + (n) if n > 1 T(n) = 2 T(n/2) + n = 2 ((n/2)log(n/2) + (n/2)) + n = n (log(n/2)) + 2n = n log n – n + 2n = n log n + n = O(n log n )

9. Comparing the Algorithms Best Average Worst Case Case Case Bubble Sort O(n) O(n2) O(n2) Insertion Sort O(n) O(n2) O(n2) Selection Sort O(n2) O(n2) O(n2) Merge Sort O(n log n) O(n log n) O(n log n) Quick Sort O(n log n) O(n log n) O(n2) Heap Sort O(n log n) O(n log n) O(n log n)

10. Thank You

Data Structure and Algorithms Merge Sort

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