Merge k Sorted Lists
You are given an array of $k$ linked-lists, each sorted in ascending order. Merge all the linked-lists into one sorted linked list. Use a Min-Heap (Priority Queue) for an efficient solution.
Why Interviewers Ask This
Amazon asks this question to evaluate a candidate's mastery of advanced data structures and ability to optimize for scalability. Specifically, they test your capacity to move beyond naive solutions that degrade performance as input size grows. The interviewer looks for proficiency in using Min-Heaps to manage complexity efficiently, demonstrating the 'Invent and Simplify' leadership principle by finding elegant algorithmic shortcuts rather than brute-force methods.
How to Answer This Question
1. Clarify constraints immediately: Ask about the number of lists (k) and average list length to determine if O(N*k) or O(N log k) is required.
2. Propose the naive approach first: Briefly explain merging two lists repeatedly, noting its O(N*k) time complexity to show you understand the baseline problem.
3. Introduce the optimal solution: Pivot to the Min-Heap strategy. Explain how inserting the head of each list into a priority queue allows constant-time access to the smallest element.
4. Detail the algorithm flow: Describe the loop where you extract the minimum node, append it to the result, and push the next node from that specific list back into the heap.
5. Analyze complexity: Explicitly state the Time Complexity as O(N log k) and Space Complexity as O(k), justifying why this meets Amazon's high-performance standards for distributed systems.
Key Points to Cover
- Demonstrating knowledge of O(N log k) vs O(N*k) time complexity trade-offs
- Correctly implementing the Min-Heap logic for dynamic element selection
- Handling null pointers and empty list edge cases gracefully
- Using a dummy node to simplify linked list construction logic
- Articulating space complexity as O(k) due to heap storage
Sample Answer
To solve the Merge k Sorted Lists problem efficiently, I would avoid the naive approach of iteratively merging pairs of lists, which results in a quadratic time complexity relative to the total number of nodes. Instead, I would leverage a Min-Heap to achieve an optimal O(N log k) runtime, where N is the total number of nodes and k is the number of lists.
First, I initialize a Min-Heap and insert the head node of every non-empty linked list into it. This heap will always maintain the smallest available value at the root. Next, I create a dummy head pointer for my result list to simplify edge case handling. I then enter a loop that continues until the heap is empty. In each iteration, I poll the smallest node from the top of the heap. I attach this node to my result list and advance my result pointer. Crucially, if the polled node has a next sibling in its original list, I insert that next node into the heap. This ensures the heap always contains the current smallest candidates from all active lists.
This approach mirrors Amazon's focus on scalable algorithms. By maintaining only k elements in the heap at any time, we ensure logarithmic insertion and extraction costs. For example, if we have three lists with 100 nodes each, we perform roughly 300 * log(3) operations rather than 9000 operations required by repeated pairwise merging. This demonstrates both technical depth and an understanding of resource optimization.
Common Mistakes to Avoid
- Failing to handle the case where one or more input lists are initially empty
- Attempting to merge lists sequentially without a heap, leading to TLE on large inputs
- Forgetting to re-insert the next node from the same list after extracting the current minimum
- Not explicitly stating the time and space complexity analysis during the solution walkthrough
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