A Simple Yet Fast Algorithm for the Closest-Pair Problem Using Sorted Projections on Multi-Dimensions

We present a simple greedy algorithm, QuickCP, for finding the closestpair of points on a plane. It is based on the observation that if two points are close to each other, then it is very likely that their sorted projections to x-axis and/or to y-axis will reflect that closeness. Therefore we order our search starting from the pairs with closest x-projections (and closest y-projections) to find the closest pair quickly and make a fast exit. Empirical data up to 226 (over sixty million points) indicates that this approach quickly detects the closest pair and it is, on average 70 percent faster than the classical divide-and-conquer algorithm registering, to the best of our knowledge, the fastest recorded solution times in the literature for the planar closest-pair problem. A second contribution of our work is that QuickCP can be used as part of the divide-and-conquer algorithms to handle small sub-problems. Measurements on data up to 226 points show that this co-operation speeds up the basic divide-andconquer algorithm on average 50 percent.