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Merge k Sorted Arrays

Medium Heaps Merge K Sorted Original on GeeksforGeeks

Merge k Sorted Arrays is a medium heaps problem solved with the merge k sorted pattern. The best approach, optimal (min-heap of array heads), runs in O(N log k) time and O(k) space. Below are 2 approaches in Java, from concatenate and sort up.

Problem

Given k sorted arrays, merge them into one sorted array.

Examples

Example 1

Input
arr = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
Output
[1, 2, 3, 4, 5, 6, 7, 8, 9]

Example 2

Input
arr = [[1, 5], [2, 3], [4, 9]]
Output
[1, 2, 3, 4, 5, 9]

Constraints

  • 1 <= k <= 100; each array has length k.
  • Each array is sorted.

Animated walkthrough

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Solutions

Try it yourself first. Then compare: each approach lists its idea, the steps, its time and space, and the Java code.

ApproachTimeSpace
Concatenate and sortO(N log N)O(N)
Optimal (min-heap of array heads)O(N log k)O(k)

1Concatenate and sort

TimeO(N log N)
SpaceO(N)

Put everything into one list and sort it.

  1. Add all elements; Collections.sort.
Java
class Solution {
    public static ArrayList<Integer> mergeKArrays(int[][] arr, int K) {
        ArrayList<Integer> out = new ArrayList<>();
        for (int[] row : arr) for (int x : row) out.add(x);
        Collections.sort(out);
        return out;
    }
}

2Optimal (min-heap of array heads)

TimeO(N log k)
SpaceO(k)Besides the output.

Keep one entry per array in a min-heap: {value, row, col}. Poll the smallest, append it, and push the next element of the same row.

  1. Offer {arr[r][0], r, 0} for each row.
  2. Poll; append; if col + 1 < len, offer the next element.
Java
class Solution {
    public static ArrayList<Integer> mergeKArrays(int[][] arr, int K) {
        PriorityQueue<int[]> heap = new PriorityQueue<>((a, b) -> Integer.compare(a[0], b[0]));
        for (int r = 0; r < arr.length; r++) if (arr[r].length > 0) heap.offer(new int[] { arr[r][0], r, 0 });
        ArrayList<Integer> out = new ArrayList<>();
        while (!heap.isEmpty()) {
            int[] e = heap.poll();
            out.add(e[0]);
            int r = e[1], c = e[2] + 1;
            if (c < arr[r].length) heap.offer(new int[] { arr[r][c], r, c });
        }
        return out;
    }
}

Edge cases to test

  • Arrays of different lengths (the heap method still works)

Hints

Hint 1

Store (value, array index, element index) in the heap so you know what to push next.

FAQ

What is the best time complexity for Merge k Sorted Arrays?

Optimal (min-heap of array heads) runs in O(N log k) time and O(k) extra space.

Which pattern does Merge k Sorted Arrays use?

It is a heaps problem that uses the merge k sorted pattern. Other problems with the same pattern: Merge k Sorted Lists.

Is there a brute force solution for Merge k Sorted Arrays?

Yes. Concatenate and sort takes O(N log N) time and O(N) space. Put everything into one list and sort it.

Which edge cases should I test for Merge k Sorted Arrays?

Arrays of different lengths (the heap method still works).