An Optimized Divide-and-Conquer Algorithm for the Closest-Pair Problem in the Planar Case

We present an engineered version of the divide-and-conquer algorithm for finding the closest pair of points, within a given set of points in the XY-plane. For this version of the algorithm we show that only two pairwise comparisons are required in the combine step, for each point that lies in the 2δ-wide vertical slab. The correctness of the algorithm is shown for all Minkowski distances with p ≥ 1. We also show empirically that, although the time complexity of the algorithm is still O(n lg n), the reduction in the total number of comparisons leads to a significant reduction in the total execution time, for inputs with size sufficiently large.