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1 Computer Programming Dr. Deepak B Phatak Dr. Supratik Chakraborty Department of Computer Science and Engineering Session: Merge Sort in C++ and Its Analysis Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 1
2 Quick Recap of Relevant Topics The sorting problem Selection sort Merge sort Intuition Divide-and-conquer strategy Key role of merging sorted sub-arrays 2 Dr. Deepak B. Phatak & Dr. Supratik Chakraborty,
3 Overview of This Lecture Merge sort C++ implementation Analysis of performance Counting basic steps to sort an array of size n 3 Dr. Deepak B. Phatak & Dr. Supratik Chakraborty,
4 Merge Sort: Basic Idea Divide Similar but simpler problem Sort first half of array Similar but simpler problem Sort second half of array Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 4
5 Sorting by Divide-and-Conquer Divide Sort Similar but simpler problem Sort second half of array Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 5
6 Sorting by Divide-and-Conquer Divide Sort Merge Sort Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 6
7 What Were The Steps? Divide an array of size n into two sub-arrays of size n/2 Sub-array sizes may differ by 1 if n is odd Easy! Sort each sub-array of size n/2 Use same technique as for sorting original array (recurse!!!) Termination case of recursion: arrays of size 1 Merge sorted sub-arrays, each of size n/2 One pass over each sorted sub-array Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 7
8 Merge Sort In C++ int main() { int i, n, A[100]; // Variable declarations cout << Give count of integers: ; cin >> n;// Read and validate inputs if ((n <= 0) (n > 100)) { cout << Invalid input! << endl; return -1; cout << Give << n << integers to sort. << endl; for (i = 0; i < n; i++) { cin >> A[i]; // Read elements of array A mergesort(a, 0, n); // Sort elements A[0] A[n-1] Rest of code return 0; Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 8
9 Merge Sort In C++ int main() { int i, n, A[100]; // Variable declarations cout << Give count of integers: ; cin >> n;// Read and validate inputs if ((n <= 0) (n > 100)) { cout << Invalid input! << endl; return -1; cout << Give << n << integers to sort. << endl; for (i = 0; i < n; i++) { cin >> A[i]; // Read elements of array A mergesort(a, 0, n); // Sort elements A[0] A[n-1] Rest of code return 0; Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 9
10 Merge Sort In C++ int main() { int i, n, A[100]; // Variable declarations cout << Give count of integers: ; cin >> n;// Read and validate inputs if ((n <= 0) (n > 100)) { cout << Invalid input! << endl; return -1; cout << Give << n << integers to sort. << endl; for (i = 0; i < n; i++) { cin >> A[i]; // Read elements of array A mergesort(a, 0, n); // Sort elements A[0] A[n-1] Rest of code return 0; Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 10
11 Merge Sort In C++ // PRECONDITION: start < end, both within array bounds of A void mergesort(int A[], int start, int end) { if (end == start +1) { return; //Subarray of interest has 1 element int mid = (start + end)/2; //Get mid-index of subarray of interest mergesort(a, start, mid); // Sort subarray A[start] A[mid-1] mergesort(a, mid, end); // Sort subarray A[mid] A[end-1] // Merge sorted subarrays A[start] A[mid-1] and A[mid] A[end-1] mergesortedsubarrays(a, start, mid, end); return; // POSTCONDITION: A[start] A[end-1] sorted in decreasing order Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 11
12 Merge Sort In C++ // PRECONDITION: start < end, both within array bounds of A void mergesort(int A[], int start, int end) { if (end == start +1) { return; //Subarray of interest has 1 element int mid = (start + end)/2; //Get mid-index of subarray of interest mergesort(a, start, mid); // Sort subarray A[start] A[mid-1] mergesort(a, mid, end); // Sort subarray A[mid] A[end-1] // Merge sorted subarrays A[start] A[mid-1] and A[mid] A[end-1] mergesortedsubarrays(a, Termination start, case mid, of end); recursion return; // POSTCONDITION: A[start] A[end-1] sorted in decreasing order Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 12
13 Merge Sort In C++ // PRECONDITION: start < end, both within array bounds of A void mergesort(int A[], int start, int end) { if (end == start +1) { return; //Subarray of interest has 1 element int mid = (start + end)/2; //Get mid-index of subarray of interest mergesort(a, start, mid); // Sort subarray A[start] A[mid-1] mergesort(a, mid, end); // Sort subarray A[mid] A[end-1] // Merge sorted subarrays A[start] A[mid-1] and A[mid] A[end-1] mergesortedsubarrays(a, start, mid, end); return; // POSTCONDITION: A[start] A[end-1] sorted in decreasing order Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 13
14 Merge Sort In C++ // PRECONDITION: start < end, both within array bounds of A void mergesort(int A[], int start, int end) { if (end == start +1) { return; //Subarray of interest has 1 element int mid = (start + end)/2; //Get mid-index of subarray of interest mergesort(a, start, mid); // Sort subarray A[start] A[mid-1] mergesort(a, mid, end); // Sort subarray A[mid] A[end-1] // Merge sorted subarrays A[start] A[mid-1] and A[mid] A[end-1] mergesortedsubarrays(a, start, mid, end); return; // POSTCONDITION: A[start] A[end-1] sorted in decreasing order Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 14
15 Merge Sort In C++ // PRECONDITION: A[start] A[mid-1] and A[mid] A[end-1] sorted in // decreasing order void mergesortedsubarrays(int A[], int start, int mid, int end) { int i, j; int tempa[100], index = start; for (i = start, j = mid; ((i < mid) (j < end)); ) { // Merging loop // Determine whether A[i] or A[j] should appear next in sorted order // Update tempa[index] accordingly index ++; // end of merging loop // Copy tempa[start] tempa[end-1] to A[start] A[end-1] return; // POSTCONDITION: A[start] A[end-1] sorted in decreasing order Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 15
16 Merge Sort In C++ // PRECONDITION: A[start] A[mid-1] and A[mid] A[end-1] sorted in // decreasing order void mergesortedsubarrays(int A[], int start, int mid, int end) { int i, j; int tempa[100], index = start; for (i = start, j = mid; ((i < mid) (j < end)); ) { // Merging loop Running index for second subarray // Determine whether A[i] or A[j] should appear next in sorted order // Update tempa[index] accordingly A[mid] A[end-1] index ++; // end of merging loop // Copy tempa[start] tempa[end-1] to A[start] A[end-1] Running index for first subarray A[start] A[mid-1] return; // POSTCONDITION: A[start] A[end-1] sorted in decreasing order Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 16
17 Merge Sort In C++ // PRECONDITION: A[start] A[mid-1] and A[mid] A[end-1] sorted in // decreasing order void mergesortedsubarrays(int A[], int start, int mid, int end) { int i, j; int tempa[100], index = start; for (i = start, j = mid; ((i < mid) (j < end)); ) { // Merging loop // Determine whether A[i] or A[j] should appear next in sorted order // Update tempa[index] accordingly index ++; // end of merging loop // Copy Merged-and-sorted tempa[start] tempa[end-1] subarrayto temporarily A[start] A[end-1] stored in return; tempa[start] tempa[end-1] // POSTCONDITION: A[start] A[end-1] sorted in decreasing order Index in tempa where next element of merged-and-sorted subarray is stored Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 17
18 Merge Sort In C++ // PRECONDITION: A[start] A[mid-1] and A[mid] A[end-1] sorted in // decreasing order void mergesortedsubarrays(int A[], int start, int mid, int end) { int i, j; int tempa[100], index = start; for (i = start, j = mid; ((i < mid) (j < end)); ) { // Merging loop // Determine whether A[i] or A[j] should appear next in sorted order // Update tempa[index] accordingly index ++; // end of merging loop // Copy tempa[start] tempa[end-1] to A[start] A[end-1] return; // POSTCONDITION: A[start] A[end-1] sorted in decreasing order Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 18
19 Merging Loop In C++ for (i = start, j = mid; ((i < mid) (j < end)); ) { // Merging loop if ((i < mid) && (j < end)) { // None of the two subarrays fully seen yet // Find if (A[j] which > A[i]) of A[i] {tempa[index] (from subarray = A[j]; A[start] j++; A[mid-1]) or // A[j] else (from {tempa[index] subarray A[mid] = A[i]; A[end-1] i++; is the next sorted element, // copy that element to tempa[index] else { if // (i One < mid) of the {tempa[index] two subarrays = A[i]; fully i++; seen. // A[mid] A[end-1] seen else // Copy if (j elements < end) {tempa[index] from the other = subarray A[j]; j++; to // tempa. A[start] A[mid-1] seen index ++; // end of merging loop Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 19
20 Merging Loop In C++ for (i = start, j = mid; ((i < mid) (j < end)); ) { // Merging loop if ((i < mid) && (j < end)) { // None of the two subarrays fully seen yet // Find if (A[j] which > A[i]) of A[i] {tempa[index] (from subarray = A[j]; A[start] j++; A[mid-1]) or // A[j] else (from {tempa[index] subarray A[mid] = A[i]; A[end-1] i++; is the next sorted element, // copy that element to tempa[index] else { if //(i One < mid) of the {tempa[index] two subarrays = A[i]; fully i++; seen. // A[mid] A[end-1] seen else // Copy if (j < elements end) {tempa[index] from the other = A[j]; subarray j++; // to A[start] tempa. A[mid-1] seen index ++; // end of merging loop Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 20
21 Merging Loop In C++ for (i = start, j = mid; ((i < mid) (j < end)); ) { // Merging loop if ((i < mid) && (j < end)) { // None of the two subarrays fully seen yet // Find if (A[j] which > A[i]) of A[i] {tempa[index] (from subarray = A[j]; A[start] j++; A[mid-1]) or // A[j] else (from {tempa[index] subarray A[mid] = A[i]; A[end-1] i++; is the next sorted element, // copy that element to tempa[index] else { if (i < mid) {tempa[index] = A[i]; i++; // A[mid] A[end-1] seen else {tempa[index] = A[j]; j++; // A[start] A[mid-1] seen index ++; // end of merging loop Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 21
22 Merging Loop In C++ for (i = start, j = mid; ((i < mid) (j < end)); ) { // Merging loop if ((i < mid) && (j < end)) { // None of the two subarrays fully seen yet // Find if (A[j] which > A[i]) of A[i] {tempa[index] (from subarray = A[j]; A[start] j++; A[mid-1]) or // A[j] else (from {tempa[index] subarray A[mid] = A[i]; A[end-1]) i++; is the next sorted element, // copy that element to tempa[index] else { if (i < mid) {tempa[index] = A[i]; i++; // A[mid] A[end-1] seen else {tempa[index] = A[j]; j++; // A[start] A[mid-1] seen index ++; // end of merging loop Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 22
23 Merging Loop In C++ for (i = start, j = mid; ((i < mid) (j < end)); ) { // Merging loop if ((i < mid) && (j < end)) { // None of the two subarrays fully seen yet if (A[j] > A[i]) {tempa[index] = A[j]; j++; else {tempa[index] = A[i]; i++; else { if (i < mid) {tempa[index] = A[i]; i++; // A[mid] A[end-1] seen else {tempa[index] = A[j]; j++; // A[start] A[mid-1] seen index ++; // end of merging loop Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 23
24 Merging Loop In C++: The Whole Story for (i = start, j = mid; ((i < mid) (j < end)); ) { // Merging loop if ((i < mid) && (j < end)) { // None of the two subarrays fully seen yet if (A[j] > A[i]) {tempa[index] = A[j]; j++; else {tempa[index] = A[i]; i++; else { if (i < mid) {tempa[index] = A[i]; i++; // A[mid] A[end-1] seen else {tempa[index] = A[j]; j++; // A[start] A[mid-1] seen index ++; // end of merging loop Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 24
25 Merge Sort In C++ // PRECONDITION: A[start] A[mid-1] and A[mid] A[end-1] sorted in // decreasing order void mergesortedsubarrays(int A[], int start, int mid, int end) { int i, j; int tempa[100], index = start; for (i = start, j = mid; ((i < mid) (j < end)); ) { // Merging loop // Determine for (i = whether start; i A[i] < end; or A[j] i++) should {A[i] appear = tempa[i]; next in sorted order // Update tempa[index] accordingly index ++; // end of merging loop // Copy tempa[start] tempa[end-1] to A[start] A[end-1] return; // POSTCONDITION: A[start] A[end-1] sorted in decreasing order Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 25
26 Basic Steps in mergesortedsubarrays Reading two array elements, comparing them, writing an element in tempa, incrementing indices Copying an element from tempa to A Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 26
27 Counting Basic Steps In mergesortedsubarrays Suppose mergesortedsubarrays called with sub-arrays A[start] A[mid-1] and A[mid] A[end-1], each of size n/2 At most n/2 basic steps to iterate over A[start] A[mid-1] At most n/2 basic steps to iterate over A[mid] A[end-1] In all, at most (n/2 + n/2) or n basic steps to get merged sorted subarray in tempa n basic steps to copy tempa[start] tempa[end-1] to A[start] A[end-1] Overall, at most 2n basic steps Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 27
28 Counting Basic Steps in Merge Sort Let T n denote maximum count of basic steps to merge sort an array of size n Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 28
29 Merge Sort: Looking Back // PRECONDITION: start < end, both within array bounds of A void mergesort(int A[], int start, int end) { if (end == start +1) { return; //Subarray of interest has 1 element int mid = (start + end)/2; //Get mid-index of subarray of interest mergesort(a, start, mid); // Sort subarray A[start] A[mid-1] mergesort(a, mid, end); // Sort subarray A[mid] A[end-1] // Merge sorted subarrays A[start] A[mid-1] and A[mid] A[end-1] mergesortedsubarrays(a, start, mid, end); return; // POSTCONDITION: A[start] A[end-1] sorted in decreasing order Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 29
30 Counting Basic Steps in Merge Sort Let Tn denote maximum count of basic steps to merge sort an array of size n T n = T n/2 + T n/2 + 2n mergesortedsubarrays Second recursive call of mergesort First recursive call of mergesort T 1 = 1 Solution of recurrence: T n 2n. log 2 n + n Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 30
31 Count of basic steps Comparing Basic Steps With Selection Sort Much faster than selection sort for large n Array size Grow as (n-1).(n+2)/2 Grow as 2n.(log n) + n 2 Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 31
32 Summary Merge sort C++ implementation using recursion Analysis of performance Counting basic steps in sorting an array of size n Clear winner performance-wise over selection sort Dr. Deepak B. Phatak & Dr. Supratik Chakraborty, 32
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