Merge K sorted arrays | Set 3 ( Using Divide and Conquer Approach )

Giving k sorted arrays, each of size N, the task is to merge them into a single sorted array.

Examples:

Input: arr[][] = {{5, 7, 15, 18},
                   {1, 8, 9, 17},
                   {1, 4, 7, 7}}
Output: {1, 1, 4, 5, 7, 7, 7, 8, 9, 15, 17, 18}

Input: arr[][] = {{3, 2, 1}
                   {6, 5, 4}}
Output:  {1, 2, 3, 4, 5, 6}

Prerequisite: Merge Sort

Simple Approach: A simple solution is to append all the arrays one after another and sort them.

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// C++ program to merge K
// sorted arrays
#include <bits/stdc++.h>
using namespace std;
#define N 4
  
// Merge and sort k arrays
void merge_and_sort(
    int* output, int arr[][N],
    int n, int k)
{
    // Put the elments in sorted array.
    for (int i = 0; i < k; i++) {
        for (int j = 0; j < n; j++) {
            output[i * n + j] = arr[i][j];
        }
    }
  
    // Sort the output array
    sort(output, output + n * k);
}
  
// Driver Function
int main()
{
    // Input 2D-array
    int arr[][N] = { { 5, 7, 15, 18 },
                     { 1, 8, 9, 17 },
                     { 1, 4, 7, 7 } };
    int n = N;
  
    // Number of arrays
    int k = sizeof(arr) / sizeof(arr[0]);
  
    // Output array
    int* output = new int[n * k];
  
    merge_and_sort(output, arr, n, k);
  
    // Print merged array
    for (int i = 0; i < n * k; i++)
        cout << output[i] << " ";
  
    return 0;
}

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Output:



1 1 4 5 7 7 7 8 9 15 17 18

Complexity Analysis:

  • Time Complexity: O(N*k*log(N*k)).
    The size of all elements is n*k so the time complexity is O(N*k * log(N*k))
  • Space Complexity: O(N*k).
    To store the output array O(N*k) space is required.

Efficient Solution:
Approach: The idea becomes clear once we start looking at the k arrays as the intermediate state of the merge sort algorithm.
Since there are k arrays that are already sorted, merge the k arrays. Create a recursive function which will take k arrays and divide them into two parts and call the function recursively with each half. The base cases are when the value of k is less than 3.
See this article to merge two arrays in O(n) time.

Algorithm:

  1. Initialize the output array with the size N*k.
  2. Call the function divide. Let l and r represent the range of arrays that are to be merged and thus vary between 0 to k-1.
  3. At each step, we call the left and right half of the range recursively so that, they will be sorted and stored in the output array.
  4. After that, we merge the left and right half. For merging, we need to determine the range of indexes for the left and right halves in the output array. We can easily find that.
    1. Left part will start from the index l * n of the output array.
    2. Similarly, right part will start from the index ((l + r) / 2 + 1) * n of the output array.

C++

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// C++ program to merge K
// sorted arrays
  
#include <iostream>
#define n 4
  
using namespace std;
  
// Function to perform merge operation
void merge(int l, int r, int* output)
{
    // to store the starting point
    // of left and right array
    int l_in = l * n, r_in
                      = ((l + r) / 2 + 1) * n;
  
    // To store the size of left and
    // right array
    int l_c = ((l + r) / 2 - l + 1) * n;
    int r_c = (r - (l + r) / 2) * n;
  
    // array to temporarily store left
    // and right array
    int l_arr[l_c], r_arr[r_c];
  
    // storing data in left array
    for (int i = 0; i < l_c; i++)
        l_arr[i] = output[l_in + i];
  
    // storing data in right array
    for (int i = 0; i < r_c; i++)
        r_arr[i] = output[r_in + i];
  
    // to store the current index of
    // temporary left and right array
    int l_curr = 0, r_curr = 0;
  
    // to store the current index
    // for output array
    int in = l_in;
  
    // two pointer merge for
    // two sorted arrays
    while (
        l_curr + r_curr < l_c + r_c) {
        if (
            r_curr == r_c
            || (l_curr != l_c
                && l_arr[l_curr] < r_arr[r_curr]))
            output[in]
                = l_arr[l_curr],
                l_curr++, in++;
        else
            output[in]
                = r_arr[r_curr],
                r_curr++, in++;
    }
}
  
// Code to drive merge-sort and
// create recursion tree
void divide(int l, int r, int* output,
            int arr[][n])
{
    if (l == r) {
  
        /* base step to initialize the output
           array before performing merge 
           operation */
        for (int i = 0; i < n; i++)
            output[l * n + i] = arr[l][i];
  
        return;
    }
  
    // To sort left half
    divide(l, (l + r) / 2,
           output, arr);
  
    // To sort right half
    divide((l + r) / 2 + 1, r,
           output, arr);
  
    // Merge the left and right half
    merge(l, r, output);
}
  
// Driver Function
int main()
{
    // input 2D-array
    int arr[][n] = { { 5, 7, 15, 18 },
                     { 1, 8, 9, 17 },
                     { 1, 4, 7, 7 } };
  
    // Number of arrays
    int k = sizeof(arr) / sizeof(arr[0]);
  
    // Output array
    int* output = new int[n * k];
  
    divide(0, k - 1, output, arr);
  
    // Print merged array
    for (int i = 0; i < n * k; i++)
        cout << output[i] << " ";
  
    return 0;
}

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Java

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// Java program to merge
// K sorted arrays
import java.util.*;
  
class GFG {
  
    static int n = 4;
  
    // Function to perform
    // merge operation
    static void merge(
        int l, int r, int[] output)
    {
        // To store the starting point
        // of left and right array
        int l_in = l * n, r_in
                          = ((l + r) / 2 + 1) * n;
  
        // to store the size of left and
        // right array
        int l_c = ((l + r) / 2 - l + 1) * n;
        int r_c = (r - (l + r) / 2) * n;
  
        // array to temporarily store left
        // and right array
        int l_arr[] = new int[l_c],
            r_arr[] = new int[r_c];
  
        // storing data in left array
        for (int i = 0; i < l_c; i++)
            l_arr[i] = output[l_in + i];
  
        // storing data in right array
        for (int i = 0; i < r_c; i++)
            r_arr[i] = output[r_in + i];
  
        // to store the current index of
        // temporary left and right array
        int l_curr = 0, r_curr = 0;
  
        // to store the current index
        // for output array
        int in = l_in;
  
        // two pointer merge for two sorted arrays
        while (l_curr + r_curr < l_c + r_c) {
            if (
                r_curr == r_c
                || (l_curr != l_c
                    && l_arr[l_curr] < r_arr[r_curr])) {
                output[in] = l_arr[l_curr];
                l_curr++;
                in++;
            }
            else {
                output[in] = r_arr[r_curr];
                r_curr++;
                in++;
            }
        }
    }
  
    // Code to drive merge-sort and
    // create recursion tree
    static void divide(int l, int r, int[] output,
                       int arr[][])
    {
        if (l == r) {
  
            /* base step to initialize the output 
        array before performing merge 
        operation */
            for (int i = 0; i < n; i++)
                output[l * n + i] = arr[l][i];
  
            return;
        }
  
        // to sort left half
        divide(l, (l + r) / 2, output, arr);
  
        // to sort right half
        divide((l + r) / 2 + 1, r, output, arr);
  
        // merge the left and right half
        merge(l, r, output);
    }
  
    // Driver Code
    public static void main(String[] args)
    {
        // input 2D-array
        int arr[][] = { { 5, 7, 15, 18 },
                        { 1, 8, 9, 17 },
                        { 1, 4, 7, 7 } };
  
        // Number of arrays
        int k = arr.length;
  
        // Output array
        int[] output = new int[n * k];
  
        divide(0, k - 1, output, arr);
  
        // Print merged array
        for (int i = 0; i < n * k; i++)
            System.out.print(output[i] + " ");
    }
}
  
/* This code contributed by PrinciRaj1992 */

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Python3

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# Python3 program to merge K sorted arrays 
n = 4 ;
  
# Function to perform merge operation 
def merge(l, r, output) :
      
    # to store the starting point of 
    # left and right array 
    l_in = l * n ;
    r_in = ((l + r) // 2 + 1) * n; 
  
    # to store the size of left and 
    # right array 
    l_c = ((l + r) // 2 - l + 1) * n; 
    r_c = (r - (l + r) // 2) * n; 
  
    # array to temporarily store left 
    # and right array 
    l_arr = [0] * l_c; r_arr = [0] * r_c; 
  
    # storing data in left array 
    for i in range(l_c) :
        l_arr[i] = output[l_in + i]; 
  
    # storing data in right array 
    for i in range(r_c) :
        r_arr[i] = output[r_in + i]; 
  
    # to store the current index of 
    # temporary left and right array 
    l_curr = 0 ; r_curr = 0
  
    # to store the current index 
    # for output array 
    in1 = l_in; 
  
    # two pointer merge for two sorted arrays 
    while (l_curr + r_curr < l_c + r_c) :
        if (r_curr == r_c or (l_curr != l_c and
            l_arr[l_curr] < r_arr[r_curr])) :
            output[in1] = l_arr[l_curr]; 
            l_curr += 1; in1 += 1
        else :
            output[in1] = r_arr[r_curr];
            r_curr += 1; in1 += 1
  
# Code to drive merge-sort and 
# create recursion tree 
def divide(l, r, output, arr) :
      
    if (l == r) : 
  
        # base step to initialize the output 
        # array before performing merge 
        # operation 
        for i in range(n) :
            output[l * n + i] = arr[l][i]; 
  
        return
      
    # to sort left half 
    divide(l, (l + r) // 2, output, arr); 
  
    # to sort right half 
    divide((l + r) // 2 + 1, r, output, arr); 
  
    # merge the left and right half 
    merge(l, r, output); 
  
# Driver code 
if __name__ == "__main__" :
  
    # input 2D-array 
    arr = [[ 5, 7, 15, 18 ], 
           [ 1, 8, 9, 17 ], 
           [ 1, 4, 7, 7 ]]; 
      
    # Number of arrays 
    k = len(arr); 
      
    # Output array
    output = [0] * (n * k); 
      
    divide(0, k - 1, output, arr); 
      
    # Print merged array 
    for i in range(n * k) :
        print(output[i], end = " "); 
  
# This code is contributed by Ryuga

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C#

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// C# program to merge K sorted arrays
using System;
  
class GFG {
  
    static int n = 4;
  
    // Function to perform merge operation
    static void merge(int l, int r, int[] output)
    {
        // to store the starting point of left
        // and right array
        int l_in = l * n, r_in = ((l + r) / 2 + 1) * n;
  
        // to store the size of left and
        // right array
        int l_c = ((l + r) / 2 - l + 1) * n;
        int r_c = (r - (l + r) / 2) * n;
  
        // array to temporarily store left
        // and right array
        int[] l_arr = new int[l_c];
        int[] r_arr = new int[r_c];
  
        // storing data in left array
        for (int i = 0; i < l_c; i++)
            l_arr[i] = output[l_in + i];
  
        // storing data in right array
        for (int i = 0; i < r_c; i++)
            r_arr[i] = output[r_in + i];
  
        // to store the current index of
        // temporary left and right array
        int l_curr = 0, r_curr = 0;
  
        // to store the current index
        // for output array
        int index = l_in;
  
        // two pointer merge for two sorted arrays
        while (l_curr + r_curr < l_c + r_c) {
            if (r_curr == r_c || (l_curr != l_c && l_arr[l_curr] < r_arr[r_curr])) {
                output[index] = l_arr[l_curr];
                l_curr++;
                index++;
            }
            else {
                output[index] = r_arr[r_curr];
                r_curr++;
                index++;
            }
        }
    }
  
    // Code to drive merge-sort and
    // create recursion tree
    static void divide(int l, int r, int[] output,
                       int[, ] arr)
    {
        if (l == r) {
  
            /* base step to initialize the output 
        array before performing merge 
        operation */
            for (int i = 0; i < n; i++)
                output[l * n + i] = arr[l, i];
  
            return;
        }
  
        // to sort left half
        divide(l, (l + r) / 2, output, arr);
  
        // to sort right half
        divide((l + r) / 2 + 1, r, output, arr);
  
        // merge the left and right half
        merge(l, r, output);
    }
  
    // Driver Code
    public static void Main(String[] args)
    {
        // input 2D-array
        int[, ] arr = { { 5, 7, 15, 18 },
                        { 1, 8, 9, 17 },
                        { 1, 4, 7, 7 } };
  
        // Number of arrays
        int k = arr.GetLength(0);
  
        // Output array
        int[] output = new int[n * k];
  
        divide(0, k - 1, output, arr);
  
        // Print merged array
        for (int i = 0; i < n * k; i++)
            Console.Write(output[i] + " ");
    }
}
  
// This code has been contributed by 29AjayKumar

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PHP

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<?php
// PHP program to merge K sorted arrays 
$n = 4; 
  
// Function to perform merge operation 
function merge($l, $r, &$output
    global $n;
      
    // to store the starting point of left 
    // and right array 
    $l_in = $l * $n;
    $r_in = ((int)(($l + $r) / 2) + 1) * $n
  
    // to store the size of left and 
    // right array 
    $l_c = (int)(((($l + $r) / 2) - $l + 1) * $n); 
    $r_c = ($r - (int)(($l + $r) / 2)) * $n
  
    // array to temporarily store left 
    // and right array 
    $l_arr = array_fill(0, $l_c, 0);
    $r_arr = array_fill(0, $r_c, 0); 
  
    // storing data in left array 
    for ($i = 0; $i < $l_c; $i++) 
        $l_arr[$i] = $output[$l_in + $i]; 
  
    // storing data in right array 
    for ($i = 0; $i < $r_c; $i++) 
        $r_arr[$i] = $output[$r_in + $i]; 
  
    // to store the current index of 
    // temporary left and right array 
    $l_curr = 0;
    $r_curr = 0; 
  
    // to store the current index 
    // for output array 
    $in = $l_in
  
    // two pointer merge for two sorted arrays 
    while ($l_curr + $r_curr < $l_c + $r_c)
    
        if ($r_curr == $r_c || ($l_curr != $l_c && 
            $l_arr[$l_curr] < $r_arr[$r_curr])) 
        {
            $output[$in] = $l_arr[$l_curr]; 
            $l_curr++; $in++; 
        }
        else
        {
            $output[$in] = $r_arr[$r_curr]; 
            $r_curr++; $in++; 
        }
    
  
// Code to drive merge-sort and 
// create recursion tree 
function divide($l, $r, &$output, $arr
    global $n;
    if ($l == $r)
    
  
        /* base step to initialize the output 
        array before performing merge 
        operation */
        for ($i = 0; $i < $n; $i++) 
            $output[$l * $n + $i] = $arr[$l][$i]; 
  
        return
    
  
    // to sort left half 
    divide($l, (int)(($l + $r) / 2), $output, $arr); 
  
    // to sort right half 
    divide((int)(($l + $r) / 2) + 1, $r, $output, $arr); 
  
    // merge the left and right half 
    merge($l, $r, $output); 
  
// Driver Code
  
// input 2D-array 
$arr = array(array( 5, 7, 15, 18 ), 
             array( 1, 8, 9, 17 ), 
             array( 1, 4, 7, 7 )); 
  
// Number of arrays 
$k = count($arr); 
  
// Output array 
$output = array_fill(0, $n * $k, 0); 
  
divide(0, $k - 1, $output, $arr); 
  
// Print merged array 
for ($i = 0; $i < $n * $k; $i++) 
        print($output[$i] . " ");
          
// This code is contributed by mits 
?>

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Output:

1 1 4 5 7 7 7 8 9 15 17 18

Complexity Analysis:

  • Time Complexity: O(N*k*log(k)).
    In each level the array of size N*k is traversed once, and the number of levels are log(k).
  • Space Complexity: O(N*k).
    To store the output array O(N*k) space is required.

We can also solve this problem by using min-heap.

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