How Can You Efficiently Merge K Sorted Lists?

Merging k sorted lists is a common problem encountered in technical interviews. This task involves taking multiple sorted lists and combining them into a single sorted list. This problem can seem complex at first, but it can be solved efficiently with the right approach.

Given k sorted lists, you can typically visualize them as arrays or linked lists. The goal is to produce a new list that maintains the sorted order after merging. There are several methods to solve this. Let’s explore a couple of the most efficient strategies.

Using a Min-Heap

One of the best approaches to solve this problem efficiently is by using a min-heap (or priority queue). The idea is to leverage the property of heaps where the minimum element can always be accessed in O(1) time. Here’s a step-by-step breakdown of how this method works:

1. Initialize a min-heap: Insert the first element of each of the k lists into the heap. Along with the value, you also need to keep track of which list the element came from and its index in that list.

2. Extract the minimum element: The root of the heap will give you the smallest element. Remove it from the heap and add it to the result list.

3. Insert the next element: After extracting the minimum, check if there's another element left in the same list from which the minimum came. If so, insert that next element into the heap.

4. Repeat: Continue this process until all elements from all lists are processed.

Code Implementation

Below is a Python implementation of this approach using the heapq module, which provides a simple way to maintain a min-heap:

PythonCopy code

Time Complexity Analysis

The time complexity of this method is O(N log k), where N is the total number of elements across all k lists. Each insertion and extraction from the min-heap takes O(log k) time, and since each of the N elements will be added and removed exactly once, the overall efficiency remains optimal.

Alternative Method: Merge Pairwise

For those unfamiliar with heaps, another simpler approach is to merge the k lists pairwise. This can be achieved through a nested loop that iteratively merges two sorted lists at a time. While this method might be easier to understand, it tends to be less efficient than the min-heap approach.

In practice, combining efficiency with clarity is essential when tackling this problem in an interview setting. Depending on your comfort level and the constraints of the problem, both methods can serve well in producing the desired merged sorted list. Experiment with both to understand their strengths and when to apply them effectively.