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Merge k sorted arrays | Set 1
  • Difficulty Level : Hard
  • Last Updated : 01 Mar, 2021
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Given k sorted arrays of size n each, merge them and print the sorted output.
Example: 
 

Input: 
k = 3, n = 4 
arr[][] = { {1, 3, 5, 7}, 
{2, 4, 6, 8}, 
{0, 9, 10, 11}} ;
Output: 0 1 2 3 4 5 6 7 8 9 10 11 
Explanation: The output array is a sorted array that contains all the elements of the input matrix. 
Input: 
k = 3, n = 4 
arr[][] = { {1, 5, 6, 8}, 
{2, 4, 10, 12}, 
{3, 7, 9, 11}, 
{13, 14, 15, 16}} ;
Output: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 
Explanation: The output array is a sorted array that contains all the elements of the input matrix. 
 

 

Naive Approach: The very naive method is to create an output array of size n * k and then copy all the elements into the output array followed by sorting.
 

  • Algorithm: 
    1. Create a output array of size n * k.
    2. Traverse the matrix from start to end and insert all the elements in output array.
    3. Sort and print the output array.
  • Implementation: 
     

CPP14




// C++ program to merge k sorted arrays of size n each.
#include<bits/stdc++.h>
using namespace std;
#define n 4
 
 
// A utility function to print array elements
void printArray(int arr[], int size)
{
   for (int i=0; i < size; i++)
       cout << arr[i] << " ";
}
 
// This function takes an array of arrays as an argument and
// All arrays are assumed to be sorted. It merges them together
// and prints the final sorted output.
void mergeKArrays(int arr[][n], int a, int output[])
{
    int c=0;
     
    //traverse the matrix
    for(int i=0; i<a; i++)
    {
        for(int j=0; j<n ;j++)
            output[c++]=arr[i][j];
    }
     
    //sort the array
    sort(output,output + n*a);
     
}
  
  
// Driver program to test above functions
int main()
{
    // Change n at the top to change number of elements
    // in an array
    int arr[][n] =  {{2, 6, 12, 34},
                     {1, 9, 20, 1000},
                     {23, 34, 90, 2000}};
    int k = sizeof(arr)/sizeof(arr[0]);
     
    int output[n*k];
     
    mergeKArrays(arr, 3, output);
  
    cout << "Merged array is " << endl;
    printArray(output, n*k);
  
    return 0;
}

Java




// Java program to merge k sorted arrays of size n each.
 
import java.io.*;
import java.util.*;
 
class GFG {
 
    // This function takes an array of arrays as an argument
    // and
    // All arrays are assumed to be sorted. It merges them
    // together and prints the final sorted output.
    public static void mergeKArrays(int[][] arr, int a,
                                    int[] output)
    {
        int c = 0;
 
        // traverse the matrix
        for (int i = 0; i < a; i++) {
            for (int j = 0; j < 4; j++)
                output[c++] = arr[i][j];
        }
 
        // sort the array
        Arrays.sort(output);
    }
 
    // A utility function to print array elements
    public static void printArray(int[] arr, int size)
    {
        for (int i = 0; i < size; i++)
            System.out.print(arr[i] + " ");
    }
   
    // Driver program to test above functions
    public static void main(String[] args)
    {
        int[][] arr = { { 2, 6, 12, 34 },
                        { 1, 9, 20, 1000 },
                        { 23, 34, 90, 2000 } };
        int k = 4;
        int n = 3;
        int[] output = new int[n * k];
 
        mergeKArrays(arr, n, output);
 
        System.out.println("Merged array is ");
        printArray(output, n * k);
    }
}

C#




// C# program to merge k sorted arrays of size n each.
using System;
public class GFG
{
 
  // This function takes an array of arrays as an argument
  // and
  // All arrays are assumed to be sorted. It merges them
  // together and prints the readonly sorted output.
  public static void mergeKArrays(int[,] arr, int a,
                                  int[] output)
  {
    int c = 0;
 
    // traverse the matrix
    for (int i = 0; i < a; i++)
    {
      for (int j = 0; j < 4; j++)
        output[c++] = arr[i,j];
    }
 
    // sort the array
    Array.Sort(output);
  }
 
  // A utility function to print array elements
  public static void printArray(int[] arr, int size)
  {
    for (int i = 0; i < size; i++)
      Console.Write(arr[i] + " ");
  }
 
  // Driver program to test above functions
  public static void Main(String[] args)
  {
    int[,] arr = { { 2, 6, 12, 34 },
                  { 1, 9, 20, 1000 },
                  { 23, 34, 90, 2000 } };
    int k = 4;
    int n = 3;
    int[] output = new int[n * k];
    mergeKArrays(arr, n, output);
    Console.WriteLine("Merged array is ");
    printArray(output, n * k);
  }
}
 
// This code is contributed by Rajput-Ji
Output: 



Merged array is 
1 2 6 9 12 20 23 34 34 90 1000 2000

 

  • Complexity Analysis: 
    • Time Complexity :O(n*k*log(n*k)). 
      since resulting array is of N*k size.
    • Space Complexity :O(n*k), The output array is of size n*k.

Efficient Approach The process might begin with merging arrays into groups of two. After the first merge, we have k/2 arrays. Again merge arrays in groups, now we have k/4 arrays. This is similar to merge sort. Divide k arrays into two halves containing an equal number of arrays until there are two arrays in a group. This is followed by merging the arrays in a bottom-up manner. 
 

  • Algorithm: 
    1. Create a recursive function which takes k arrays and returns the output array.
    2. In the recursive function, if the value of k is 1 then return the array else if the value of k is 2 then merge the two arrays in linear time and return the array.
    3. If the value of k is greater than 2 then divide the group of k elements into two equal halves and recursively call the function, i.e 0 to k/2 array in one recursive function and k/2 to k array in another recursive function.
    4. Print the output array.
  • Implementation: 
     

CPP14




// C++ program to merge k sorted arrays of size n each.
#include<bits/stdc++.h>
using namespace std;
#define n 4
 
// Merge arr1[0..n1-1] and arr2[0..n2-1] into
// arr3[0..n1+n2-1]
void mergeArrays(int arr1[], int arr2[], int n1,
                             int n2, int arr3[])
{
    int i = 0, j = 0, k = 0;
   
    // Traverse both array
    while (i<n1 && j <n2)
    {
        // Check if current element of first
        // array is smaller than current element
        // of second array. If yes, store first
        // array element and increment first array
        // index. Otherwise do same with second array
        if (arr1[i] < arr2[j])
            arr3[k++] = arr1[i++];
        else
            arr3[k++] = arr2[j++];
    }
   
    // Store remaining elements of first array
    while (i < n1)
        arr3[k++] = arr1[i++];
   
    // Store remaining elements of second array
    while (j < n2)
        arr3[k++] = arr2[j++];
}
 
// A utility function to print array elements
void printArray(int arr[], int size)
{
   for (int i=0; i < size; i++)
       cout << arr[i] << " ";
}
 
// This function takes an array of arrays as an argument and
// All arrays are assumed to be sorted. It merges them together
// and prints the final sorted output.
void mergeKArrays(int arr[][n],int i,int j, int output[])
{
    //if one array is in range
    if(i==j)
    {
        for(int p=0; p < n; p++)
        output[p]=arr[i][p];
        return;
    }
     
    //if only two arrays are left them merge them
    if(j-i==1)
    {
        mergeArrays(arr[i],arr[j],n,n,output);
        return;
    }
     
    //output arrays
    int out1[n*(((i+j)/2)-i+1)],out2[n*(j-((i+j)/2))];
     
    //divide the array into halves
    mergeKArrays(arr,i,(i+j)/2,out1);
    mergeKArrays(arr,(i+j)/2+1,j,out2);
     
    //merge the output array
    mergeArrays(out1,out2,n*(((i+j)/2)-i+1),n*(j-((i+j)/2)),output);
     
}
  
  
// Driver program to test above functions
int main()
{
    // Change n at the top to change number of elements
    // in an array
    int arr[][n] =  {{2, 6, 12, 34},
                     {1, 9, 20, 1000},
                     {23, 34, 90, 2000}};
    int k = sizeof(arr)/sizeof(arr[0]);
    int output[n*k];
    mergeKArrays(arr,0,2, output);
  
    cout << "Merged array is " << endl;
    printArray(output, n*k);
  
    return 0;
}

Java




// Java program to merge k sorted arrays of size n each.
import java.util.*;
 
class GFG{
  static final int n = 4;
 
  // Merge arr1[0..n1-1] and arr2[0..n2-1] into
  // arr3[0..n1+n2-1]
  static void mergeArrays(int arr1[], int arr2[], int n1,
                          int n2, int arr3[])
  {
    int i = 0, j = 0, k = 0;
 
    // Traverse both array
    while (i<n1 && j <n2)
    {
      // Check if current element of first
      // array is smaller than current element
      // of second array. If yes, store first
      // array element and increment first array
      // index. Otherwise do same with second array
      if (arr1[i] < arr2[j])
        arr3[k++] = arr1[i++];
      else
        arr3[k++] = arr2[j++];
    }
 
    // Store remaining elements of first array
    while (i < n1)
      arr3[k++] = arr1[i++];
 
    // Store remaining elements of second array
    while (j < n2)
      arr3[k++] = arr2[j++];
  }
 
  // A utility function to print array elements
  static void printArray(int arr[], int size)
  {
    for (int i = 0; i < size; i++)
      System.out.print(arr[i]+  " ");
  }
 
  // This function takes an array of arrays as an argument and
  // All arrays are assumed to be sorted. It merges them together
  // and prints the final sorted output.
  static void mergeKArrays(int arr[][], int i, int j, int output[])
  {
    // if one array is in range
    if(i == j)
    {
      for(int p = 0; p < n; p++)
        output[p] = arr[i][p];
      return;
    }
 
    //if only two arrays are left them merge them
    if(j - i == 1)
    {
      mergeArrays(arr[i], arr[j], n, n, output);
      return;
    }
 
    //output arrays
    int []out1 = new int[n*(((i + j) / 2) - i + 1)];
    int []out2 = new int[n*(j - ((i + j) / 2))];
 
    //divide the array into halves
    mergeKArrays(arr, i, (i + j) / 2, out1);
    mergeKArrays(arr, (i + j) / 2 + 1, j, out2);
 
    //merge the output array
    mergeArrays(out1, out2, n * (((i + j) / 2) - i + 1), n * (j - ((i + j) / 2)), output);
  }
 
 
  // Driver program to test above functions
  public static void main(String[] args)
  {
 
    // Change n at the top to change number of elements
    // in an array
    int arr[][] =  {{2, 6, 12, 34},
                    {1, 9, 20, 1000},
                    {23, 34, 90, 2000}};
    int k = arr.length;
    int []output = new int[n*k];
    mergeKArrays(arr,0,2, output);
 
    System.out.print("Merged array is " +"\n");
    printArray(output, n*k);
  }
}
 
// This code is contributed by gauravrajput1
  •  
Output: 
Merged array is 
1 2 6 9 12 20 23 34 34 90 1000 2000

 

  •  
  • Complexity Analysis: 
    • Time Complexity: O( n * k * log k). 
      There are log k levels as in each level the k arrays are divided in half and at each level the k arrays are traversed. So time Complexity is O( n * k ).
    • Space Complexity: O( n * k * log k). 
      In each level O( n*k ) space is required So Space Complexity is O( n * k * log k).

 

 

Alternative Efficient Approach: The idea is to use Min Heap. This MinHeap based solution has the same time complexity which is O(NK log K). But for a different and particular sized array, this solution works much better. The process must start with creating a MinHeap and inserting the first element of all the k arrays. Remove the root element of Minheap and put it in the output array and insert the next element from the array of removed element. To get the result the step must continue until there is no element in the MinHeap left. 
MinHeap: A Min-Heap is a complete binary tree in which the value in each internal node is smaller than or equal to the values in the children of that node. Mapping the elements of a heap into an array is trivial: if a node is stored at index k, then its left child is stored at index 2k + 1 and its right child at index 2k + 2.
 

  • Algorithm: 
    1. Create a min Heap and insert the first element of all k arrays.
    2. Run a loop until the size of MinHeap is greater than zero.
    3. Remove the top element of the MinHeap and print the element.
    4. Now insert the next element from the same array in which the removed element belonged.
    5. If the array doesn’t have any more elements, then replace root with infinite.After replacing the root, heapify the tree.
  • Implementation: 
     

C++




// C++ program to merge k sorted
// arrays of size n each.
#include<bits/stdc++.h>
using namespace std;
 
#define n 4
 
// A min-heap node
struct MinHeapNode
{
// The element to be stored
    int element;
 
// index of the array from which the element is taken
    int i;
 
// index of the next element to be picked from the array
    int j;
};
 
// Prototype of a utility function to swap two min-heap nodes
void swap(MinHeapNode *x, MinHeapNode *y);
 
// A class for Min Heap
class MinHeap
{
 
// pointer to array of elements in heap
    MinHeapNode *harr;
 
// size of min heap
    int heap_size;
public:
    // Constructor: creates a min heap of given size
    MinHeap(MinHeapNode a[], int size);
 
    // to heapify a subtree with root at given index
    void MinHeapify(int );
 
    // to get index of left child of node at index i
    int left(int i) { return (2*i + 1); }
 
    // to get index of right child of node at index i
    int right(int i) { return (2*i + 2); }
 
    // to get the root
    MinHeapNode getMin() { return harr[0]; }
 
    // to replace root with new node x and heapify() new root
    void replaceMin(MinHeapNode x) { harr[0] = x;  MinHeapify(0); }
};
 
// This function takes an array of arrays as an argument and
// All arrays are assumed to be sorted. It merges them together
// and prints the final sorted output.
int *mergeKArrays(int arr[][n], int k)
{
 
// To store output array
    int *output = new int[n*k]; 
 
    // Create a min heap with k heap nodes.
    // Every heap node has first element of an array
    MinHeapNode *harr = new MinHeapNode[k];
    for (int i = 0; i < k; i++)
    {
 
// Store the first element
        harr[i].element = arr[i][0];
 
// index of array
        harr[i].i = i;
 
 // Index of next element to be stored from the array
        harr[i].j = 1;
    }
 
// Create the heap
    MinHeap hp(harr, k);
 
    // Now one by one get the minimum element from min
    // heap and replace it with next element of its array
    for (int count = 0; count < n*k; count++)
    {
        // Get the minimum element and store it in output
        MinHeapNode root = hp.getMin();
        output[count] = root.element;
 
        // Find the next elelement that will replace current
        // root of heap. The next element belongs to same
        // array as the current root.
        if (root.j < n)
        {
            root.element = arr[root.i][root.j];
            root.j += 1;
        }
        // If root was the last element of its array
// INT_MAX is for infinite       
else root.element =  INT_MAX;
 
        // Replace root with next element of array
        hp.replaceMin(root);
    }
 
    return output;
}
 
// FOLLOWING ARE IMPLEMENTATIONS OF
// STANDARD MIN HEAP METHODS FROM CORMEN BOOK
// Constructor: Builds a heap from a given
// array a[] of given size
MinHeap::MinHeap(MinHeapNode a[], int size)
{
    heap_size = size;
    harr = a;  // store address of array
    int i = (heap_size - 1)/2;
    while (i >= 0)
    {
        MinHeapify(i);
        i--;
    }
}
 
// A recursive method to heapify a
// subtree with root at given index.
// This method assumes that the subtrees
// are already heapified
void MinHeap::MinHeapify(int i)
{
    int l = left(i);
    int r = right(i);
    int smallest = i;
    if (l < heap_size && harr[l].element < harr[i].element)
        smallest = l;
    if (r < heap_size && harr[r].element < harr[smallest].element)
        smallest = r;
    if (smallest != i)
    {
        swap(&harr[i], &harr[smallest]);
        MinHeapify(smallest);
    }
}
 
// A utility function to swap two elements
void swap(MinHeapNode *x, MinHeapNode *y)
{
    MinHeapNode temp = *x;  *x = *y;  *y = temp;
}
 
// A utility function to print array elements
void printArray(int arr[], int size)
{
   for (int i=0; i < size; i++)
       cout << arr[i] << " ";
}
 
// Driver program to test above functions
int main()
{
    // Change n at the top to change number of elements
    // in an array
    int arr[][n] =  {{2, 6, 12, 34},
                     {1, 9, 20, 1000},
                     {23, 34, 90, 2000}};
    int k = sizeof(arr)/sizeof(arr[0]);
 
    int *output = mergeKArrays(arr, k);
 
    cout << "Merged array is " << endl;
    printArray(output, n*k);
 
    return 0;
}

Java




// Java program to merge k sorted
// arrays of size n each.
 
// A min heap node
class MinHeapNode
{
    int element; // The element to be stored
     
     // index of the array from
     // which the element is taken
    int i;
     
    // index of the next element
    // to be picked from array
    int j;
 
    public MinHeapNode(int element, int i, int j)
    {
        this.element = element;
        this.i = i;
        this.j = j;
    }
};
 
// A class for Min Heap
class MinHeap
{
    MinHeapNode[] harr; // Array of elements in heap
    int heap_size; // Current number of elements in min heap
 
    // Constructor: Builds a heap from
    // a given array a[] of given size
    public MinHeap(MinHeapNode a[], int size)
    {
        heap_size = size;
        harr = a;
        int i = (heap_size - 1)/2;
        while (i >= 0)
        {
            MinHeapify(i);
            i--;
        }
    }
 
    // A recursive method to heapify a subtree
    // with the root at given index This method
    // assumes that the subtrees are already heapified
    void MinHeapify(int i)
    {
        int l = left(i);
        int r = right(i);
        int smallest = i;
        if (l < heap_size && harr[l].element < harr[i].element)
            smallest = l;
        if (r < heap_size && harr[r].element < harr[smallest].element)
            smallest = r;
        if (smallest != i)
        {
            swap(harr, i, smallest);
            MinHeapify(smallest);
        }
    }
 
    // to get index of left child of node at index i
    int left(int i) { return (2*i + 1); }
 
    // to get index of right child of node at index i
    int right(int i) { return (2*i + 2); }
 
    // to get the root
    MinHeapNode getMin()
    {
        if(heap_size <= 0)
        {
            System.out.println("Heap underflow");
            return null;
        }
        return harr[0];
    }
 
    // to replace root with new node
    // "root" and heapify() new root
    void replaceMin(MinHeapNode root) {
        harr[0] = root;
        MinHeapify(0);
    }
 
    // A utility function to swap two min heap nodes
    void swap(MinHeapNode[] arr, int i, int j) {
        MinHeapNode temp = arr[i];
        arr[i] = arr[j];
        arr[j] = temp;
    }
 
    // A utility function to print array elements
    static void printArray(int[] arr) {
        for(int i : arr)
            System.out.print(i + " ");
        System.out.println();
    }
 
    // This function takes an array of
    // arrays as an argument and All
    // arrays are assumed to be sorted.
    // It merges them together and
    // prints the final sorted output.
    static void mergeKSortedArrays(int[][] arr, int k)
    {
        MinHeapNode[] hArr = new MinHeapNode[k];
        int resultSize = 0;
        for(int i = 0; i < arr.length; i++)
        {
            MinHeapNode node = new MinHeapNode(arr[i][0],i,1);
            hArr[i] = node;
            resultSize += arr[i].length;
        }
 
        // Create a min heap with k heap nodes. Every heap node
        // has first element of an array
        MinHeap mh = new MinHeap(hArr, k);
 
        int[] result = new int[resultSize];     // To store output array
 
        // Now one by one get the minimum element from min
        // heap and replace it with next element of its array
        for(int i = 0; i < resultSize; i++)
        {
 
            // Get the minimum element and store it in result
            MinHeapNode root = mh.getMin();
            result[i] = root.element;
 
            // Find the next element that will replace current
            // root of heap. The next element belongs to same
            // array as the current root.
            if(root.j < arr[root.i].length)
                root.element = arr[root.i][root.j++];
            // If root was the last element of its array
            else
                root.element = Integer.MAX_VALUE;
 
            // Replace root with next element of array
            mh.replaceMin(root);
        }
 
        printArray(result);
 
    }
 
    // Driver code
    public static void main(String args[]){
        int[][] arr= {{2, 6, 12, 34},
                {1, 9, 20, 1000},
                {23, 34, 90, 2000}};
 
        System.out.println("Merged array is :");
 
        mergeKSortedArrays(arr,arr.length);
    }
};
 
// This code is contributed by shubham96301

C#




// C# program to merge k sorted
// arrays of size n each.
using System;
 
// A min heap node
public class MinHeapNode
{
    public int element; // The element to be stored
     
    // index of the array from
    // which the element is taken
    public int i;
     
    // index of the next element
    // to be picked from array
    public int j;
 
    public MinHeapNode(int element, int i, int j)
    {
        this.element = element;
        this.i = i;
        this.j = j;
    }
};
 
// A class for Min Heap
public class MinHeap
{
    MinHeapNode[] harr; // Array of elements in heap
    int heap_size; // Current number of elements in min heap
 
    // Constructor: Builds a heap from
    // a given array a[] of given size
    public MinHeap(MinHeapNode []a, int size)
    {
        heap_size = size;
        harr = a;
        int i = (heap_size - 1) / 2;
        while (i >= 0)
        {
            MinHeapify(i);
            i--;
        }
    }
 
    // A recursive method to heapify a subtree
    // with the root at given index This method
    // assumes that the subtrees are already heapified
    void MinHeapify(int i)
    {
        int l = left(i);
        int r = right(i);
        int smallest = i;
        if (l < heap_size &&
            harr[l].element < harr[i].element)
            smallest = l;
        if (r < heap_size &&
            harr[r].element < harr[smallest].element)
            smallest = r;
        if (smallest != i)
        {
            swap(harr, i, smallest);
            MinHeapify(smallest);
        }
    }
 
    // to get index of left child of node at index i
    int left(int i) { return (2 * i + 1); }
 
    // to get index of right child of node at index i
    int right(int i) { return (2 * i + 2); }
 
    // to get the root
    MinHeapNode getMin()
    {
        if(heap_size <= 0)
        {
            Console.WriteLine("Heap underflow");
            return null;
        }
        return harr[0];
    }
 
    // to replace root with new node
    // "root" and heapify() new root
    void replaceMin(MinHeapNode root)
    {
        harr[0] = root;
        MinHeapify(0);
    }
 
    // A utility function to swap two min heap nodes
    void swap(MinHeapNode[] arr, int i, int j)
    {
        MinHeapNode temp = arr[i];
        arr[i] = arr[j];
        arr[j] = temp;
    }
 
    // A utility function to print array elements
    static void printArray(int[] arr)
    {
        foreach(int i in arr)
            Console.Write(i + " ");
        Console.WriteLine();
    }
 
    // This function takes an array of
    // arrays as an argument and All
    // arrays are assumed to be sorted.
    // It merges them together and
    // prints the final sorted output.
    static void mergeKSortedArrays(int[,] arr, int k)
    {
        MinHeapNode[] hArr = new MinHeapNode[k];
        int resultSize = 0;
        for(int i = 0; i < arr.GetLength(0); i++)
        {
            MinHeapNode node = new MinHeapNode(arr[i, 0], i, 1);
            hArr[i] = node;
            resultSize += arr.GetLength(1);
        }
 
        // Create a min heap with k heap nodes.
        // Every heap node has first element of an array
        MinHeap mh = new MinHeap(hArr, k);
 
        int[] result = new int[resultSize];     // To store output array
 
        // Now one by one get the minimum element
        // from min heap and replace it with
        // next element of its array
        for(int i = 0; i < resultSize; i++)
        {
 
            // Get the minimum element and
            // store it in result
            MinHeapNode root = mh.getMin();
            result[i] = root.element;
 
            // Find the next element that will
            // replace current root of heap.
            // The next element belongs to same
            // array as the current root.
            if(root.j < arr.GetLength(1))
                root.element = arr[root.i,root.j++];
                 
            // If root was the last element of its array
            else
                root.element = int.MaxValue;
 
            // Replace root with next element of array
            mh.replaceMin(root);
        }
        printArray(result);
    }
 
    // Driver code
    public static void Main(String []args)
    {
        int[,] arr = {{2, 6, 12, 34},
                      {1, 9, 20, 1000},
                      {23, 34, 90, 2000}};
 
        Console.WriteLine("Merged array is :");
 
        mergeKSortedArrays(arr, arr.GetLength(0));
    }
};
 
// This code is contributed by 29AjayKumar
  •  
Output: 
Merged array is 
1 2 6 9 12 20 23 34 34 90 1000 2000

 

  •  
  • Complexity Analysis: 
    • Time Complexity :O( n * k * log k), Insertion and deletion in a Min Heap requires log k time. So the Overall time compelxity is O( n * k * log k)
    • Space Complexity :O(k), If Output is not stored then the only space required is the Min-Heap of k elements. So space Comeplexity is O(k).

Merge k sorted arrays | Set 2 (Different Sized Arrays)
Thanks to vignesh for suggesting this problem and initial solution. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
 

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