The 'do better than full sort' constraint is the whole problem.
Use binary search on the value range to find the k-th smallest element, counting how many elements are ≤ mid in O(n+m) time. Alternatively, use a min-heap to merge the sorted rows and extract k elements. Discuss trade-offs and optimize for the given constraints.
Pro tip: Clarify constraints upfront (e.g., matrix size, k range) to choose the best approach. Mention that the binary search method is O((n+m) log(max-min)) and the heap method is O(k log n), and pick based on expected k.
Ask about matrix dimensions, value ranges, and whether k is 1-indexed. Discuss edge cases like k=1, k=n*m, or empty matrix.
Explain that you can binary search the answer between the smallest and largest elements. For each mid, count elements ≤ mid in O(n+m) by starting from the bottom-left corner.
Describe how to count elements ≤ mid: start at bottom-left, move up if current > mid, else move right and add (current row index + 1) to count. This leverages row and column sorting.
Mention using a min-heap to merge rows: push first element of each row, then pop and push next from same row k times. Compare time and space complexity with binary search.
State time and space complexities for both methods. Recommend binary search for large k or when value range is small, and heap for small k. Code the chosen solution.
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