MergeSort. Merging two arrays. Merging Two Sorted Arrays

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1 MergeSort This is a much more efficient sorting technique than the bubble Sort and the insertion Sort at least in terms of speed. Although the bubble and insertion sorts take O(N2) time, the mergesort is O(N*logN). Merging Two Sorted Arrays The heart of the mergesort algorithm is the merging of two already sorted arrays. Merging two sorted arrays A and B creates a third array, C, that contains all the elements of A and B, also arranged in sorted order. We ll examine the merging process first; later we ll see how it s used in sorting. Imagine two sorted arrays. They don t need to be the same size. Let s say array A has 4 elements and array B has 6 elements. They will be merged into an array C that starts with 10 empty cells. In the below figure, the circled numbers indicate the order in which elements are transferred from A and B to C. The below table shows the comparisons necessary to determine which element will be copied. The steps in the table correspond to the steps in the figure. Following each comparison, the smaller element is copied to A. Merging two arrays

2 MERGING OPERATIONS Notice that because B is empty following step 8, no more comparisons are necessary; all the remaining elements are simply copied from A into C.

3 THE merge.cpp PROGRAM //merge.cpp //demonstrates merging two arrays into a third #include <iostream> using namespace std; // void merge( int[], int, int[], int, int;) ][ void display(int[], int); //prototypes // int main)( { int arraya[] = {23, 47, 81, 95}; //source A int arrayb[] = {7, 14, 39, 55, 62, 74}; //source B int arrayc[10]; //destination merge(arraya, 4, arrayb, 6, arrayc); //merge A+B-->C display(arrayc, 10); //display result return 0; } //end main)( // void merge( int arraya[], int sizea, //merge A and B into C int arrayb[], int sizeb, int arrayc ][ ) { int adex=0, bdex=0, cdex=0; while(adex < sizea && bdex < sizeb) //neither array empty if( arraya[adex] < arrayb[bdex]) arrayc[cdex++] = arraya[adex++]; else arrayc[cdex++] = arrayb[bdex++]; while(adex < sizea) //arrayb is empty, arrayc[cdex++] = arraya[adex++]; //but arraya isn t while(bdex < sizeb) //arraya is empty,

4 arrayc[cdex++] = arrayb[bdex++]; //but arrayb isn t } //end merge)( // void display(int thearray[], int size) //display array { for(int j=0; j<size; j++) cout << thearray[j] << ; cout << endl; } In main() the arrays arraya, arrayb, and arrayc are created; then the merge() member function is called to merge arraya and arrayb into arrayc, and the resulting contents of arrayc are displayed. Here s the output: The merge() function has three while loops: The first steps along both arraya and arrayb, comparing elements and copying the smaller of the two into arrayc. The second while loop deals with the situation when all the elements have been transferred out of arrayb, but arraya still has remaining elements. The loop simply copies the remaining elements from arraya into arrayc. The third loop handles the similar situation when all the elements have been transferred out of arraya, but arrayb still has remaining elements; they are copied to arrayc.

5 Sorting by Merging The idea in the mergesort is to divide an array in half, sort each half, and then use the merge() member function to merge the two halves into a single sorted array. How do you sort each half? Divide the half into two quarters, sort each of the quarters, and merge them to make a sorted half. Similarly each pair of 8ths is merged to make a sorted quarter; each pair of 16ths is merged to make a sorted 8th, and so on. You divide the array again and again until you reach a subarray with only one element. This is the base case; it s assumed an array with one element is already sorted. In mergesort() the range is divided in half each time this member function calls itself, and each time it returns it merges two smaller ranges into a larger one. As mergesort() returns from finding two arrays of one element each, it merges them into a sorted array of two elements. Each pair of resulting 2-element arrays is then merged into a 4-element array. This process continues with larger and larger arrays until the entire array is sorted. This is easiest to see when the original array size is a power of 2, as shown in the following figure.

6 Merging larger and larger arrays

7 Array size not a power of 2

8 Mergesort Algorithm and Implementation The mergesort is a divide-and-conquer algorithm: 1. Divides the array near its midpoint 2. Sorts the two half-arrays by recursive calls 3. Merges the two sorted half-arrays to obtain a fully sorted array Function implementation: void mergesort(datatype data[], int n) { if (n > 1) { int n1 = n/2; // Size of first subarray int n2 = n-n1; // Size of second subarray mergesort(data,n1); // Sort the first subarray mergesort(data+n1,n2); // Sort the second subarray merge(data,n1,n2); // Merge the 2 sorted subarrays } } Illustrating Mergesort

9 The Merge Function // Pre: data array has at least n1+n2 elements // first n1 elements of data are sorted in ascending order // next n2 elements of data are also sorted // Post: n1+n2 elements of data are rearranged in ascending order void merge(datatype data[], int n1, int n2) { DataType *temp; // Temporary array allocated dynamically int i = 0; // Index of first sorted subarray int j = n1; // Index of second sorted subarray int k = 0; // Index of merged temporary array temp = new DataType[n1+n2]; while (i < n1 && j < n1+n2) if (data[i] <= data[j]) temp[k++] = data[i++]; else temp[k++] = data[j++]; while (i < n1) temp[k++] = data[i++]; while (j < n1+n2) temp[k++] = data[j++]; for (i=0; i<n1+n2; i++) data[i] = temp[i]; delete [] temp; }

10 Illustrating the Merge Function Analysis of Mergesort To merge two subarrays into an array of n elements Approximately n comparisons are made The merge function has a time complexity = O(n) For an array of n elements The number of levels of recursive calls = O(log2 n) Similarly, the number of levels of merges = O(log2 n) Mergesort has a time complexity of O(n log2 n) in all cases A drawback of mergesort Requires a temporary array for merging two subarrays Allocating, deleting, and copying a dynamic array increases the runtime Mergesort is useful when sorting an external file The external file need not fit in memory, but can be sorted as pieces Sequential access can be used to merge two sorted files

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