Closest Pair of Points

The closest pair of points problem is a fundamental challenge in computational geometry: given `n` points in a metric space, find the pair of points with the smallest distance between them. A naive brute-force approach would be to calculate the distance between every pair of points, which would take O(n^2) time. A much more efficient solution can be achieved using a divide-and-conquer strategy, bringing the complexity down to O(n log n). The algorithm works as follows: 1. Sort the points based on their x-coordinates. 2. Divide the set of points into two equal-sized subsets by a vertical line. 3. Recursively find the closest pair in the left subset and the right subset. Let the minimum distance found so far be `d`. 4. The crucial step is to find if there's a closer pair of points where one point is in the left subset and the other is in the right. Such a pair must lie within a 'strip' of width `2d` centered around the dividing line. 5. The algorithm then considers only the points within this strip. It can be proven that for each point in the strip, we only need to check a constant number of subsequent points (sorted by y-coordinate) to find a potentially closer pair. This linear-time merge step is what makes the overall algorithm efficient.