I knew the brute force immediately, two nested loops, done.
Start by clarifying the problem constraints and expected output, then propose the divide-and-conquer algorithm that runs in O(n log n) time. Explain the algorithm step-by-step, emphasizing the merge step and the constant number of distance checks, and discuss trade-offs with simpler O(n^2) approaches.
Pro tip: Mention that the divide-and-conquer approach can be adapted to return the closest pair itself, not just the distance, and that it's the standard solution for large-scale nearest-neighbor problems in ML pipelines.
Ask about input size, whether points are unique, and if the output should be the distance or the pair. Confirm that 20,000 points is large enough to rule out O(n^2) for production.
Describe the brute-force O(n^2) approach and explain why it's inefficient for 20,000 points (400 million comparisons). This sets up the need for a better algorithm.
Explain sorting by x-coordinate, recursively finding the minimum distance in left and right halves, and then the merge step that checks points within a strip of width 2δ around the dividing line.
State that the algorithm runs in O(n log n) time due to sorting and recursive merging, and O(n) space for auxiliary arrays. Compare with O(n^2) to highlight efficiency.
Mention handling of duplicate points, floating-point precision, and how the algorithm can be parallelized or used in ML contexts like clustering or nearest-neighbor search.
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