3-way Merge Sort

• Difficulty Level : Medium
• Last Updated : 28 Jun, 2021

Prerequisite – Merge Sort

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  : 45, -2, -45, 78, 30, -42, 10, 19 , 73, 93
Output : -45 -42 -2 10 19 30 45 73 78 93

Input  : 23, -19
Output : -19  23

C++

 // C++ Program to perform 3 way Merge Sort#include 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(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 alements 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);    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 alements 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] + " ");    }}

Output:
After 3 way merge sort:
-45 -42 -2 10 19 30 45 73 78 93

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.

Similar article :
3 way Quick Sort

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