Smallest Range Covering Elements from K Lists
You have $k$ lists of sorted integers. Find the smallest range that includes at least one number from each of the $k$ lists. Use a Min-Heap.
Why Interviewers Ask This
Spotify engineers ask this to evaluate your ability to manage dynamic state across multiple data streams efficiently. It tests if you can move beyond brute-force solutions to optimize time complexity using a Min-Heap. They specifically look for your understanding of how to balance range minimization with the constraint of covering every list simultaneously.
How to Answer This Question
1. Clarify constraints: Confirm if lists are sorted and the value range, as Spotify values precise problem scoping before coding. 2. Define the strategy: Propose using a Min-Heap to track the current minimum element from each of the k lists while maintaining a running maximum. 3. Initialize: Explain pushing the first element of every list into the heap and tracking the initial max value. 4. Iterate: Describe the loop where you extract the min, update the smallest range if the current window is better, then insert the next element from that same list's source. 5. Terminate: State that the process stops when any list is exhausted, ensuring no valid range exists beyond that point. 6. Analyze: Conclude by calculating O(N log K) time complexity, where N is total elements, demonstrating optimization awareness.
Key Points to Cover
- Explicitly mention using a Min-Heap to efficiently retrieve the smallest element among k lists
- Demonstrate understanding that the range is defined by the current minimum and the running maximum
- Explain the termination condition clearly: stopping when any list is fully traversed
- Correctly derive the time complexity as O(N log K) rather than O(N*K)
- Show step-by-step logic for updating the heap and range bounds during iteration
Sample Answer
To solve the Smallest Range Covering Elements from K Lists problem, I would first verify that the input lists are indeed sorted, which allows us to leverage their order for efficiency. My approach involves a sliding window technique implemented via a Min-Heap. I will initialize a Min-Heap containing the first element from each of the k lists. Simultaneously, I'll track the maximum value currently in the heap to define the upper bound of our potential range.
In the main loop, I extract the minimum element from the heap. This element represents the lower bound of the current range. I compare the difference between the current maximum and this minimum against my best-known range. If it is smaller, I update the optimal range boundaries. Next, I identify which list this minimum came from and push the next element from that specific list into the heap. If that list is exhausted, we stop, as no further ranges can include an element from all lists.
For example, if we have lists [[1, 10], [4, 11], [7, 12]], we start with 1, 4, and 7. The range is [1, 7]. We pop 1, push 10. Now the heap has 4, 7, 10. The max is 10, min is 4. Range [4, 10] is wider than [1, 7], so we keep the original. We continue until one list runs out. This ensures an O(N log K) solution, optimizing space and time compared to checking all combinations, which aligns well with Spotify's focus on scalable data processing.
Common Mistakes to Avoid
- Attempting a brute-force solution that checks every possible combination, leading to TLE (Time Limit Exceeded)
- Failing to maintain a separate variable for the current maximum value inside the heap, causing incorrect range calculations
- Not handling edge cases where lists might be empty or contain duplicate values correctly
- Confusing the heap operations and removing elements without properly inserting the next element from the same source list
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