Brute force is O(n^2) and I knew that wasn't the answer they wanted.
Start by clarifying the problem constraints (e.g., 2D/3D, number of points, expected time complexity) and then present the divide-and-conquer algorithm that sorts points by x-coordinate, recursively finds the minimum distance in each half, and checks a strip around the dividing line. Emphasize the O(n log n) time complexity and contrast it with the brute-force O(n^2) approach.
Pro tip: Mention that in practice, for large datasets, a sweep-line algorithm or spatial hashing can be more efficient, but the divide-and-conquer approach is the classic optimal solution for this problem. Also, note that the algorithm can be adapted for high-dimensional spaces with appropriate data structures.
Ask about the dimensionality of the points, the expected number of points, and whether the distance metric is Euclidean. Confirm if the points are static or dynamic.
Mention the naive O(n^2) approach of comparing all pairs, which is simple but inefficient for large n. This sets the stage for optimization.
Explain sorting by x-coordinate, recursively solving left and right halves, and then checking the strip of width 2*delta around the median. Detail how to efficiently check the strip by sorting by y-coordinate and comparing each point to at most 7 subsequent points.
Show that the recurrence T(n) = 2T(n/2) + O(n) leads to O(n log n) time. Mention that space complexity is O(n) due to auxiliary arrays.
Mention adaptations for higher dimensions, use of sweep-line or spatial hashing for streaming data, and potential parallelization for very large datasets.
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