Counting Unit Circular Arc Intersections

TL;DR

This paper introduces a significantly faster algorithm for counting intersections among same-radius circular arcs in the plane, improving the long-standing computational complexity barrier.

Contribution

The authors develop a new algorithm that reduces the complexity of counting arc intersections from over 30 years old bounds to near-optimal performance, with adaptable complexity based on the number of intersections.

Findings

New algorithm achieves $O(n^{4/3} ext{log}^{16/3}n)$ time complexity.

Improved bounds for cases with fewer intersections, $O(n^{1+ extepsilon}+K^{1/3}n^{2/3}(rac{n^2}{n+K})^{ extepsilon} ext{log}^{16/3}n)$.

Significantly advances the computational approach to geometric intersection problems.

Abstract

Given a set of $n$ circular arcs of the same radius in the plane, we consider the problem of computing the number of intersections among the arcs. The problem was studied before and the previously best algorithm solves the problem in $O(n^{4/3+\epsilon})$ time [Agarwal, Pellegrini, and Sharir, SIAM J. Comput., 1993], for any constant $\epsilon>0$. No progress has been made on the problem for more than 30 years. We present a new algorithm of $O(n^{4/3}\log^{16/3}n)$ time and improve it to $O(n^{1+\epsilon}+K^{1/3}n^{2/3}(\frac{n^2}{n+K})^{\epsilon}\log^{16/3}n)$ time for small $K$, where $K$ is the number of intersections of all arcs.