# 3-way Merge Sort

• Difficulty Level : Hard
• Last Updated : 01 Aug, 2022

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 ``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 Code``int` `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 Sort``import` `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 Sort``using` `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

 ``

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.

Similar article: 3 way Quick Sort

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