# 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}
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Prerequisite: Merge Sort

Simple Solution: A simple solution is to append all the arrays one after another and sort them. The time complexity of this operation will be O(N*k * log(N*k)).

Efficient Solution: The idea becomes clear once we start looking at the k arrays as the intermediate state of merge sort algorithm.

Since, we have got k arrays that are already sorted, all we need to do is merge the k arrays. Merging them, will create a recursion tree of log(k) height because, at each step, the number of remaining arrays becomes half. At each height, the algorithm will take a time of O(N*k). Thus, the time complexity boils down to O( N*k * log(k) ).

Algorithm:

1. Initialize the output array with the size N*k.
2. Call the function divide. Let and 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.
• Left part will start from the index l * n of the output array.
• Similarly, right part will start from the index ((l + r) / 2 + 1) * n of the output array.

We, also have to determine the length of the left and right parts.
Then, we create two temporary arrays, ‘l_arr’ and ‘r_arr’ to store the values from the left and right part. Finally, we use two pointer technique to merge the results of the left and right part as we do in merge sort.

Below is the C++ implementation of the above algorithm:

 `// C++ program to merge K sorted arrays ` ` `  `#include ` `#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 Cunction ` `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; ` `} `

Output:

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

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

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