I knew what a sliding window was, sure, but maintaining a running median as elements enter and leave the window is a completely different problem.
Start by clarifying the problem constraints (e.g., array size, k, data types) and then discuss the brute-force approach (O(n*k log k)) before optimizing with a data structure that supports efficient insertion, deletion, and median retrieval, such as two heaps or a balanced BST. Explain the trade-offs between different approaches and finally walk through a concrete example to validate the solution.
Pro tip: Mention that for large datasets, using two heaps (max-heap for lower half, min-heap for upper half) with lazy deletion can achieve O(n log k) time, but be prepared to discuss how to handle deletions from the heaps efficiently, as that's a common follow-up.
Ask about input size, whether k is always valid, if the array can contain duplicates, and if the median definition for even-sized windows (average of two middle elements) is expected.
Explain that for each window, you could sort the elements and find the median, resulting in O(n*k log k) time. This sets a baseline for optimization.
Introduce using two heaps (max-heap for lower half, min-heap for upper half) to maintain the window's elements, allowing O(log k) insertion and deletion, and O(1) median retrieval.
Describe how to add the new element and remove the oldest element from the heaps, ensuring the heaps remain balanced and the median is correctly computed after each slide.
State the overall time complexity O(n log k) and space O(k). Discuss edge cases like k=1, k=n, and even/odd window sizes.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.