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

TL;DR

This paper introduces an optimized divide-and-conquer algorithm for the planar closest-pair problem that reduces comparisons and improves execution time while maintaining the same theoretical complexity.

Contribution

It presents a version of the algorithm requiring only two comparisons per point in the combine step, enhancing efficiency in practice.

Findings

Fewer pairwise comparisons in the combine step.

Significant reduction in total execution time for large inputs.

Algorithm remains O(n log n) in time complexity.

Abstract

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 2 delta-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.