K-Way Merge Coding Pattern

1. Introduction

The K-Way Merge pattern is a common coding technique used to merge K sorted arrays or lists into a single sorted array efficiently.

Problem Statement:

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

This pattern is widely used in:

External sorting (large datasets not fitting into memory)
Merge operations in divide and conquer algorithms
Priority queues and streaming data
Problems like merging K sorted linked lists or arrays

Example:

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

2. When to Use K-Way Merge Pattern

Use this pattern when:

You have multiple sorted arrays or lists.
You want to merge them into a single sorted array efficiently.
You are working with streaming or external data that is already partially sorted.
Sorting everything together is inefficient due to time or memory constraints.

Problem Indicators:

“Merge K sorted arrays/lists”
“Merge K sorted linked lists”
“Find the Kth smallest/largest element among sorted arrays”
“External sorting of large files”

3. Core Idea

The main idea is to use a min-heap (priority queue) to always pick the smallest (or largest) element among the K arrays.

Steps:

Initialize a min-heap that stores elements along with information about which array they came from and their index.
Insert the first element of each array into the heap.
While the heap is not empty:
- Extract the smallest element from the heap.
- Add it to the merged output.
- Insert the next element from the same array into the heap (if it exists).

This ensures the merged array remains sorted.

4. Java Implementation – Merge K Sorted Arrays

import java.util.*;

class Element {
    int value;
    int arrayIndex;
    int elementIndex;
    
    public Element(int value, int arrayIndex, int elementIndex) {
        this.value = value;
        this.arrayIndex = arrayIndex;
        this.elementIndex = elementIndex;
    }
}

public class KWayMerge {
    public static List<Integer> mergeKSortedArrays(List<List<Integer>> arrays) {
        List<Integer> result = new ArrayList<>();
        
        PriorityQueue<Element> minHeap = new PriorityQueue<>(
            Comparator.comparingInt(e -> e.value)
        );
        
        // Step 1: Insert first element of each array into min-heap
        for (int i = 0; i < arrays.size(); i++) {
            if (!arrays.get(i).isEmpty()) {
                minHeap.offer(new Element(arrays.get(i).get(0), i, 0));
            }
        }
        
        // Step 2: Extract min and insert next element from same array
        while (!minHeap.isEmpty()) {
            Element current = minHeap.poll();
            result.add(current.value);
            
            int nextIndex = current.elementIndex + 1;
            if (nextIndex < arrays.get(current.arrayIndex).size()) {
                minHeap.offer(new Element(
                    arrays.get(current.arrayIndex).get(nextIndex),
                    current.arrayIndex,
                    nextIndex
                ));
            }
        }
        
        return result;
    }
    
    public static void main(String[] args) {
        List<List<Integer>> arrays = Arrays.asList(
            Arrays.asList(1,4,7),
            Arrays.asList(2,5,8),
            Arrays.asList(3,6,9)
        );
        System.out.println(mergeKSortedArrays(arrays));
    }
}

Output: [1, 2, 3, 4, 5, 6, 7, 8, 9]

Explanation:

Min-heap always gives the current smallest element among K arrays.
Time Complexity: O(N log K), where N = total elements, K = number of arrays
Space Complexity: O(K) for the heap

5. Merge K Sorted Linked Lists

This is a variant of the same pattern.

Steps:

Use a min-heap of size K to store the head nodes of all lists.
Extract the smallest node from heap and attach to result list.
Push the next node of the extracted node into the heap.
Repeat until heap is empty.

Time Complexity: O(N log K), Space Complexity: O(K)

6. Divide & Conquer Approach

Another approach to merge K sorted arrays:

Pair arrays and merge them recursively:
- Merge arrays[0] & arrays[1] → array A
- Merge arrays[2] & arrays[3] → array B
- Merge A & B → result
Repeat until only one array remains

Time Complexity: O(N log K)

Similar to heap approach but uses recursive merging

7. Top K Problems Using K-Way Merge

The K-Way Merge pattern can also be adapted for Top K elements problems, for example:

Find K smallest numbers across K sorted arrays
Find K largest numbers in streaming data
Use min-heap of size K while merging instead of full merge

8. Tips and Observations

Always store array index and element index in heap for tracking.
Use min-heap for ascending merge, max-heap for descending.
Divide & Conquer approach is preferred for memory-sensitive environments.
Use this pattern for merging large external datasets that cannot fit into memory.
Can be extended to streams where each array/list is an infinite or very large stream.

9. Common Edge Cases

Some arrays may be empty → skip them
Arrays with duplicate values → works naturally
K = 0 → return empty list
Unequal length arrays → works naturally

10. Key Takeaways

The K-Way Merge pattern efficiently merges K sorted arrays or lists using a min-heap.
Core idea: always extract the smallest element and insert the next element from the same array.
Time Complexity: O(N log K), where N is total elements, K is number of arrays.
Space Complexity: O(K) for heap storage.
Variants:
- Merge K sorted linked lists
- Find Top K elements across K arrays
- Streaming Top K elements