Detecting all regular polygons in a point set

TL;DR

This paper presents a probabilistic algorithm for efficiently detecting all regular polygons in a point set, improving upon previous deterministic methods by combining line sweeping and random sampling techniques.

Contribution

It introduces a new algorithm that finds all regular polygons with high probability in expected sub-quadratic time, depending on the parameter alpha.

Findings

Algorithm finds all regular polygons with high probability in O(n^{2+alpha+epsilon}) time.

The method combines line sweeping for small polygons and random sampling for larger ones.

Results improve upon previous deterministic algorithms in terms of expected runtime.

Abstract

In this paper, we analyze the time complexity of finding regular polygons in a set of n points. We combine two different approaches to find regular polygons, depending on their number of edges. Our result depends on the parameter alpha, which has been used to bound the maximum number of isosceles triangles that can be formed by n points. This bound has been expressed as O(n^{2+2alpha+epsilon}), and the current best value for alpha is ~0.068. Our algorithm finds polygons with O(n^alpha) edges by sweeping a line through the set of points, while larger polygons are found by random sampling. We can find all regular polygons with high probability in O(n^{2+alpha+epsilon}) expected time for every positive epsilon. This compares well to the O(n^{2+2alpha+epsilon}) deterministic algorithm of Brass.