Merge sort involves recursively splitting the array into 2 parts, sorting and finally merging them. A variant of merge sort is called 3-way merge sort where instead of splitting the array into 2 parts we split it into 3 parts.
Merge sort recursively breaks down the arrays to subarrays of size half. Similarly, 3-way Merge sort breaks down the arrays to subarrays of size one third.
Examples:
Input: arr = [45, -2, -45, 78, 30, -42, 10, 19, 73, 93]
Output: [-45, -42, -2, 10, 19, 30, 45, 73, 78, 93]Input: arr = [23, -19]
Output: [-19, 23]
C++
// C++ Program to perform 3 way Merge Sort#include <bits/stdc++.h>using namespace std;/* Merge the sorted ranges [low, mid1), [mid1,mid2) and [mid2, high) mid1 is first midpoint index in overall range to merge mid2 is second midpoint index in overall range to merge*/void merge(vector<int> &gArray, int low, int mid1, int mid2, int high, vector<int> &destArray) { int i = low, j = mid1, k = mid2, l = low; // Choose smaller of the smallest in the three ranges while ((i < mid1) && (j < mid2) && (k < high)) { if(gArray[i] < gArray[j]) { if(gArray[i] < gArray[k]) { destArray[l++] = gArray[i++]; } else { destArray[l++] = gArray[k++]; } } else { if(gArray[j] < gArray[k]) { destArray[l++] = gArray[j++]; } else { destArray[l++] = gArray[k++]; } } } // Case where first and second ranges // have remaining values while ((i < mid1) && (j < mid2)) { if(gArray[i] < gArray[j]) { destArray[l++] = gArray[i++]; } else { destArray[l++] = gArray[j++]; } } // case where second and third ranges // have remaining values while ((j < mid2) && (k < high)) { if(gArray[j] < gArray[k]) { destArray[l++] = gArray[j++]; } else { destArray[l++] = gArray[k++]; } } // Case where first and third ranges have // remaining values while ((i < mid1) && (k < high)) { if(gArray[i] < gArray[k]) { destArray[l++] = gArray[i++]; } else { destArray[l++] = gArray[k++]; } } // Copy remaining values from the first range while (i < mid1) destArray[l++] = gArray[i++]; // Copy remaining values from the second range while (j < mid2) destArray[l++] = gArray[j++]; // Copy remaining values from the third range while (k < high) destArray[l++] = gArray[k++]; } /* Performing the merge sort algorithm on the given array of values in the rangeof indices [low, high). low is minimum index, high is maximum index (exclusive) */void mergeSort3WayRec(vector<int> &gArray, int low, int high, vector<int> &destArray) { // If array size is 1 then do nothing if (high - low < 2) return; // Splitting array into 3 parts int mid1 = low + ((high - low) / 3); int mid2 = low + 2 * ((high - low) / 3) + 1; // Sorting 3 arrays recursively mergeSort3WayRec(destArray, low, mid1, gArray); mergeSort3WayRec(destArray, mid1, mid2, gArray); mergeSort3WayRec(destArray, mid2, high, gArray); // Merging the sorted arrays merge(destArray, low, mid1, mid2, high, gArray); }void mergeSort3Way(vector<int> &gArray, int n) { // if array size is zero return null if (n == 0) return; // creating duplicate of given array vector<int> fArray(n); // copying elements of given array into // duplicate array for (int i = 0; i < n; i++) fArray[i] = gArray[i]; // sort function mergeSort3WayRec(fArray, 0, n, gArray); // copy back elements of duplicate array // to given array for (int i = 0; i < n; i++) gArray[i] = fArray[i]; } // Driver Codeint main(){ vector<int> data = {45, -2, -45, 78, 30, -42, 10, 19, 73, 93}; mergeSort3Way(data,10); cout << "After 3 way merge sort: "; for (int i = 0; i < 10; i++) { cout << data[i] << " "; } return 0;}// This code is contributed by Rashmi Kumari |
Java
// Java program to perform 3 way Merge Sortimport java.util.*;public class MergeSort3Way{ // Function for 3-way merge sort process public static void mergeSort3Way(Integer[] gArray) { // if array of size is zero returns null if (gArray == null) return; // creating duplicate of given array Integer[] fArray = new Integer[gArray.length]; // copying elements of given array into // duplicate array for (int i = 0; i < fArray.length; i++) fArray[i] = gArray[i]; // sort function mergeSort3WayRec(fArray, 0, gArray.length, gArray); // copy back elements of duplicate array // to given array for (int i = 0; i < fArray.length; i++) gArray[i] = fArray[i]; } /* Performing the merge sort algorithm on the given array of values in the rangeof indices [low, high). low is minimum index, high is maximum index (exclusive) */ public static void mergeSort3WayRec(Integer[] gArray, int low, int high, Integer[] destArray) { // If array size is 1 then do nothing if (high - low < 2) return; // Splitting array into 3 parts int mid1 = low + ((high - low) / 3); int mid2 = low + 2 * ((high - low) / 3) + 1; // Sorting 3 arrays recursively mergeSort3WayRec(destArray, low, mid1, gArray); mergeSort3WayRec(destArray, mid1, mid2, gArray); mergeSort3WayRec(destArray, mid2, high, gArray); // Merging the sorted arrays merge(destArray, low, mid1, mid2, high, gArray); } /* Merge the sorted ranges [low, mid1), [mid1, mid2) and [mid2, high) mid1 is first midpoint index in overall range to merge mid2 is second midpoint index in overall range to merge*/ public static void merge(Integer[] gArray, int low, int mid1, int mid2, int high, Integer[] destArray) { int i = low, j = mid1, k = mid2, l = low; // choose smaller of the smallest in the three ranges while ((i < mid1) && (j < mid2) && (k < high)) { if (gArray[i].compareTo(gArray[j]) < 0) { if (gArray[i].compareTo(gArray[k]) < 0) destArray[l++] = gArray[i++]; else destArray[l++] = gArray[k++]; } else { if (gArray[j].compareTo(gArray[k]) < 0) destArray[l++] = gArray[j++]; else destArray[l++] = gArray[k++]; } } // case where first and second ranges have // remaining values while ((i < mid1) && (j < mid2)) { if (gArray[i].compareTo(gArray[j]) < 0) destArray[l++] = gArray[i++]; else destArray[l++] = gArray[j++]; } // case where second and third ranges have // remaining values while ((j < mid2) && (k < high)) { if (gArray[j].compareTo(gArray[k]) < 0) destArray[l++] = gArray[j++]; else destArray[l++] = gArray[k++]; } // case where first and third ranges have // remaining values while ((i < mid1) && (k < high)) { if (gArray[i].compareTo(gArray[k]) < 0) destArray[l++] = gArray[i++]; else destArray[l++] = gArray[k++]; } // copy remaining values from the first range while (i < mid1) destArray[l++] = gArray[i++]; // copy remaining values from the second range while (j < mid2) destArray[l++] = gArray[j++]; // copy remaining values from the third range while (k < high) destArray[l++] = gArray[k++]; } // Driver function public static void main(String args[]) { // test case of values Integer[] data = new Integer[] {45, -2, -45, 78, 30, -42, 10, 19, 73, 93}; mergeSort3Way(data); System.out.println("After 3 way merge sort: "); for (int i = 0; i < data.length; i++) System.out.print(data[i] + " "); }} |
C#
// C# program to perform 3 way Merge Sortusing System;public class MergeSort3Way{ // Function for 3-way merge sort process public static void mergeSort3Way(int[] gArray) { // if array of size is zero returns null if (gArray == null) return; // creating duplicate of given array int[] fArray = new int[gArray.Length]; // copying elements of given array into // duplicate array for (int i = 0; i < fArray.Length; i++) fArray[i] = gArray[i]; // sort function mergeSort3WayRec(fArray, 0, gArray.Length, gArray); // copy back elements of duplicate array // to given array for (int i = 0; i < fArray.Length; i++) gArray[i] = fArray[i]; } /* Performing the merge sort algorithm on the given array of values in the rangeof indices [low, high). low is minimum index, high is maximum index (exclusive) */ public static void mergeSort3WayRec(int[] gArray, int low, int high, int[] destArray) { // If array size is 1 then do nothing if (high - low < 2) return; // Splitting array into 3 parts int mid1 = low + ((high - low) / 3); int mid2 = low + 2 * ((high - low) / 3) + 1; // Sorting 3 arrays recursively mergeSort3WayRec(destArray, low, mid1, gArray); mergeSort3WayRec(destArray, mid1, mid2, gArray); mergeSort3WayRec(destArray, mid2, high, gArray); // Merging the sorted arrays merge(destArray, low, mid1, mid2, high, gArray); } /* Merge the sorted ranges [low, mid1), [mid1, mid2) and [mid2, high) mid1 is first midpoint index in overall range to merge mid2 is second midpoint index in overall range to merge*/ public static void merge(int[] gArray, int low, int mid1, int mid2, int high, int[] destArray) { int i = low, j = mid1, k = mid2, l = low; // choose smaller of the smallest in the three ranges while ((i < mid1) && (j < mid2) && (k < high)) { if (gArray[i].CompareTo(gArray[j]) < 0) { if (gArray[i].CompareTo(gArray[k]) < 0) destArray[l++] = gArray[i++]; else destArray[l++] = gArray[k++]; } else { if (gArray[j].CompareTo(gArray[k]) < 0) destArray[l++] = gArray[j++]; else destArray[l++] = gArray[k++]; } } // case where first and second ranges have // remaining values while ((i < mid1) && (j < mid2)) { if (gArray[i].CompareTo(gArray[j]) < 0) destArray[l++] = gArray[i++]; else destArray[l++] = gArray[j++]; } // case where second and third ranges have // remaining values while ((j < mid2) && (k < high)) { if (gArray[j].CompareTo(gArray[k]) < 0) destArray[l++] = gArray[j++]; else destArray[l++] = gArray[k++]; } // case where first and third ranges have // remaining values while ((i < mid1) && (k < high)) { if (gArray[i].CompareTo(gArray[k]) < 0) destArray[l++] = gArray[i++]; else destArray[l++] = gArray[k++]; } // copy remaining values from the first range while (i < mid1) destArray[l++] = gArray[i++]; // copy remaining values from the second range while (j < mid2) destArray[l++] = gArray[j++]; // copy remaining values from the third range while (k < high) destArray[l++] = gArray[k++]; } // Driver function public static void Main() { // test case of values int[] data = new int[] {45, -2, -45, 78, 30, -42, 10, 19, 73, 93}; mergeSort3Way(data); Console.Write("After 3 way merge sort: "); for (int i = 0; i < data.Length; i++) Console.Write(data[i] + " "); }}// This code is contributed by saurabh_jaiswal. |
Javascript
<script>// JavaScript Program to perform 3 way Merge Sort/* Merge the sorted ranges [low, mid1), [mid1,mid2)and [mid2, high) mid1 is first midpointindex in overall range to merge mid2 is secondmidpoint index in overall range to merge*/function merge(gArray, low, mid1,mid2, high, destArray){ let i = low, j = mid1, k = mid2, l = low; // choose smaller of the smallest in the three ranges while ((i < mid1) && (j < mid2) && (k < high)) { if(gArray[i] < gArray[j]) { if(gArray[i] < gArray[k]) { destArray[l++] = gArray[i++]; } else { destArray[l++] = gArray[k++]; } } else { if(gArray[j] < gArray[k]) { destArray[l++] = gArray[j++]; } else { destArray[l++] = gArray[k++]; } } } // case where first and second ranges // have remaining values while ((i < mid1) && (j < mid2)) { if(gArray[i] < gArray[j]) { destArray[l++] = gArray[i++]; } else { destArray[l++] = gArray[j++]; } } // case where second and third ranges // have remaining values while ((j < mid2) && (k < high)) { if(gArray[j] < gArray[k]) { destArray[l++] = gArray[j++]; } else { destArray[l++] = gArray[k++]; } } // case where first and third ranges have // remaining values while ((i < mid1) && (k < high)) { if(gArray[i] < gArray[k]) { destArray[l++] = gArray[i++]; } else { destArray[l++] = gArray[k++]; } } // copy remaining values from the first range while (i < mid1) destArray[l++] = gArray[i++]; // copy remaining values from the second range while (j < mid2) destArray[l++] = gArray[j++]; // copy remaining values from the third range while (k < high) destArray[l++] = gArray[k++];}/* Performing the merge sort algorithm on thegiven array of values in the rangeof indices[low, high). low is minimum index, high ismaximum index (exclusive) */function mergeSort3WayRec(gArray, low,high, destArray){ // If array size is 1 then do nothing if (high - low < 2) return; // Splitting array into 3 parts let mid1 = low + Math.floor((high - low) / 3); let mid2 = low + 2 * Math.floor((high - low) / 3) + 1; // Sorting 3 arrays recursively mergeSort3WayRec(destArray, low, mid1, gArray); mergeSort3WayRec(destArray, mid1, mid2, gArray); mergeSort3WayRec(destArray, mid2, high, gArray); // Merging the sorted arrays merge(destArray, low, mid1, mid2, high, gArray);}function mergeSort3Way(gArray,n){ // if array size is zero return null if (n == 0) return; // creating duplicate of given array let fArray = new Array(n); // copying elements of given array into // duplicate array for (let i = 0; i < n; i++) fArray[i] = gArray[i]; // sort function mergeSort3WayRec(fArray, 0, n, gArray); // copy back elements of duplicate array // to given array for (let i = 0; i < n; i++) gArray[i] = fArray[i];}// Driver Codelet data = [45, -2, -45, 78, 30, -42, 10, 19, 73, 93];mergeSort3Way(data,10);document.write("After 3 way merge sort: ","</br>");for (let i = 0; i < 10; i++){ document.write(data[i]," ");}// This code is contributed by shinjanpatra</script> |
Python3
# Python Program to perform 3 way Merge Sort""" Merge the sorted ranges [low, mid1), [mid1,mid2) and [mid2, high) mid1 is first midpoint index in overall range to merge mid2 is second midpoint index in overall range to merge"""def merge(gArray, low, mid1, mid2, high, destArray): i = low j = mid1 k = mid2 l = low # Choose smaller of the smallest in the three ranges while ((i < mid1) and (j < mid2) and (k < high)): if(gArray[i] < gArray[j]): if(gArray[i] < gArray[k]): destArray[l] = gArray[i] l += 1 i += 1 else: destArray[l] = gArray[k] l += 1 k += 1 else: if(gArray[j] < gArray[k]): destArray[l] = gArray[j] l += 1 j += 1 else: destArray[l] = gArray[k] l += 1 k += 1 # Case where first and second ranges # have remaining values while ((i < mid1) and (j < mid2)): if(gArray[i] < gArray[j]): destArray[l] = gArray[i] l += 1 i += 1 else: destArray[l] = gArray[j] l += 1 j += 1 # case where second and third ranges # have remaining values while ((j < mid2) and (k < high)): if(gArray[j] < gArray[k]): destArray[l] = gArray[j] l += 1 j += 1 else: destArray[l] = gArray[k] l += 1 k += 1 # Case where first and third ranges have # remaining values while ((i < mid1) and (k < high)): if(gArray[i] < gArray[k]): destArray[l] = gArray[i] l += 1 i += 1 else: destArray[l] = gArray[k] l += 1 k += 1 # Copy remaining values from the first range while (i < mid1): destArray[l] = gArray[i] l += 1 i += 1 # Copy remaining values from the second range while (j < mid2): destArray[l] = gArray[j] l += 1 j += 1 # Copy remaining values from the third range while (k < high): destArray[l] = gArray[k] l += 1 k += 1""" Performing the merge sort algorithm on the given array of values in the rangeof indices [low, high). low is minimum index, high is maximum index (exclusive) """def mergeSort3WayRec(gArray, low, high, destArray): # If array size is 1 then do nothing if (high - low < 2): return # Splitting array into 3 parts mid1 = low + ((high - low) // 3) mid2 = low + 2 * ((high - low) // 3) + 1 # Sorting 3 arrays recursively mergeSort3WayRec(destArray, low, mid1, gArray) mergeSort3WayRec(destArray, mid1, mid2, gArray) mergeSort3WayRec(destArray, mid2, high, gArray) # Merging the sorted arrays merge(destArray, low, mid1, mid2, high, gArray)def mergeSort3Way(gArray, n): # if array size is zero return null if (n == 0): return # creating duplicate of given array fArray = [] # copying elements of given array into # duplicate array fArray = gArray.copy() # sort function mergeSort3WayRec(fArray, 0, n, gArray) # copy back elements of duplicate array # to given array gArray = fArray.copy() # return the sorted array return gArraydata = [45, -2, -45, 78, 30, -42, 10, 19, 73, 93]data = mergeSort3Way(data, 10)print("After 3 way merge sort: ", end="")for i in range(10): print(f"{data[i]} ", end="")# This code is contributed by Susobhan Akhuli |
C
// C Program to perform 3 way Merge Sort#include <stdio.h>/* Merge the sorted ranges [low, mid1), [mid1,mid2) and [mid2, high) mid1 is first midpoint index in overall range to merge mid2 is second midpoint index in overall range to merge*/void merge(int gArray[], int low, int mid1, int mid2, int high, int destArray[]) { int i = low, j = mid1, k = mid2, l = low; // Choose smaller of the smallest in the three ranges while ((i < mid1) && (j < mid2) && (k < high)){ if(gArray[i] < gArray[j]){ if(gArray[i] < gArray[k]) destArray[l++] = gArray[i++]; else destArray[l++] = gArray[k++]; } else{ if(gArray[j] < gArray[k]) destArray[l++] = gArray[j++]; else destArray[l++] = gArray[k++]; } } // Case where first and second ranges // have remaining values while ((i < mid1) && (j < mid2)) { if(gArray[i] < gArray[j]) destArray[l++] = gArray[i++]; else destArray[l++] = gArray[j++]; } // case where second and third ranges // have remaining values while ((j < mid2) && (k < high)) { if(gArray[j] < gArray[k]) destArray[l++] = gArray[j++]; else destArray[l++] = gArray[k++]; } // Case where first and third ranges have // remaining values while ((i < mid1) && (k < high)) { if(gArray[i] < gArray[k]) destArray[l++] = gArray[i++]; else destArray[l++] = gArray[k++]; } // Copy remaining values from the first range while (i < mid1) destArray[l++] = gArray[i++]; // Copy remaining values from the second range while (j < mid2) destArray[l++] = gArray[j++]; // Copy remaining values from the third range while (k < high) destArray[l++] = gArray[k++]; } /* Performing the merge sort algorithm on the given array of values in the rangeof indices [low, high). low is minimum index, high is maximum index (exclusive) */void mergeSort3WayRec(int gArray[], int low, int high, int destArray[]) { // If array size is 1 then do nothing if (high - low < 2) return; // Splitting array into 3 parts int mid1 = low + ((high - low) / 3); int mid2 = low + 2 * ((high - low) / 3) + 1; // Sorting 3 arrays recursively mergeSort3WayRec(destArray, low, mid1, gArray); mergeSort3WayRec(destArray, mid1, mid2, gArray); mergeSort3WayRec(destArray, mid2, high, gArray); // Merging the sorted arrays merge(destArray, low, mid1, mid2, high, gArray); }void mergeSort3Way(int gArray[], int n){ // if array size is zero return null if (n == 0) return; // creating duplicate of given array int fArray[n]; // copying elements of given array into // duplicate array for (int i = 0; i < n; i++) fArray[i] = gArray[i]; // sort function mergeSort3WayRec(fArray, 0, n, gArray); // copy back elements of duplicate array // to given array for (int i = 0; i < n; i++) gArray[i] = fArray[i]; } // Driver Codeint main(){ int data[] = {45, -2, -45, 78, 30, -42, 10, 19, 73, 93}; mergeSort3Way(data,10); printf("After 3 way merge sort: "); for (int i = 0; i < 10; i++) printf("%d ",data[i]); return 0;}// This code is contributed by Susobhan Akhuli |
Output
After 3 way merge sort: -45 -42 -2 10 19 30 45 73 78 93
How the above code works?
Here, we first copy the contents of data array to another array called fArray. Then, sort the array by finding midpoints that divide the array into 3 parts and called sort function on each array respectively. The base case of recursion is when size of array is 1 and it returns from the function. Then merging of arrays starts and finally the sorted array will be in fArray which is copied back to gArray.
Time Complexity: In case of 2-way Merge sort we get the equation: T(n) = 2T(n/2) + O(n)
Similarly, in case of 3-way Merge sort we get the equation: T(n) = 3T(n/3) + O(n)
By solving it using Master method, we get its complexity as O(n log 3n).
Although time complexity looks less compared to 2 way merge sort, the time taken actually may become higher because number of comparisons in merge function go higher. Please refer Why is Binary Search preferred over Ternary Search? for details.
Auxiliary Space Complexity: The space complexity of 3-way merge sort is same as 2-way merge sort: O(n)
Similar article: 3 way Quick Sort
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