The Question

Merge K Sorted Lists

You are given an array of $k$ linked-lists lists, where each linked-list is sorted in ascending order. Your task is to merge all the linked-lists into a single sorted linked-list and return its head. Example: Input: lists = [[1,4,5],[1,3,4],[2,6]] Output: [1,1,2,3,4,4,5,6] Constraints: - k == lists.length - 0 <= k <= 10^4 - 0 <= lists[i].length <= 500 - -10^4 <= lists[i][j] <= 10^4 - The total number of nodes in all lists does not exceed $10^4$.

Java

Priority Queue

Linked List

K-Way Merge

Questions & Insights

Clarifying Questions

What is the maximum value for

k

(number of lists) and

N

(total number of nodes across all lists)?

Can the lists be empty, or can the input array lists itself be null?

Are the node values restricted to a specific range (e.g., 32-bit integers)?

Is there a preference for space complexity? (e.g.,

O(k)

extra space for a heap vs.

O(1)

extra space using iterative divide and conquer).

Assumptions:

k

can be up to

10^4

, and

N

can be up to

10^4

The input array or individual lists within it can be null or empty.

Node values are standard integers.

We aim for an optimal time complexity of

O(N \log k)

Thinking Process

Intuition: Since each list is already sorted, the smallest overall element must be at the head of one of the

k

lists. This is a classic "k-way merge" problem.

Efficiency: Comparing all

k

heads every time we pick a node results in

O(N \cdot k)

time. To optimize, we need a data structure that can provide the minimum of

k

elements in

O(\log k)

time. A Min-Priority Queue (Min-Heap) is the standard tool for this.

Execution:

Initialize a Min-Heap and insert the head node of every non-empty list.

Use a dummy head node to build the result list conveniently.

Repeatedly extract the minimum node from the heap, attach it to the result list, and if that node has a "next" pointer, insert the next node into the heap.

Edge Cases: Handle null input, empty arrays, and lists that contain only null elements.

Implementation Breakdown

Problem Set

Functional Requirements: Merge

k

sorted linked lists into a single sorted linked list.

Constraints:

0 \le k \le 10^4

0 \le \text{total nodes} \le 10^4

Time Complexity:

O(N \log k)

Space Complexity:

O(k)

(for the heap) or

O(1)

(if using iterative divide and conquer, excluding the output).

Approach

Algorithm: K-Way Merge using Min-Heap.

Data Structure: PriorityQueue (Min-Heap) and Linked List.

Complexity:

Time:

O(N \log k)

, where

N

is the total number of nodes and

k

is the number of lists. Each node is added and removed from the heap once.

Space:

O(k)

to store the head of each list in the priority queue.

Implementation

Wrap Up

Advanced Topics

Divide and Conquer: Instead of a heap, we can merge lists in pairs (1 with 2, 3 with 4, etc.) iteratively. This also yields

O(N \log k)

time but can reduce auxiliary space to

O(1)

if done carefully without recursion.

Memory Management: In a production environment with massive lists that don't fit in RAM, this problem evolves into External Merge Sort. We would read chunks of lists from disk, merge them, and write back to disk.

Thread Safety: If the lists are being produced by different threads, we could use a PriorityBlockingQueue or a producer-consumer pattern to merge streams of data in real-time.