Counting Unit Circular Arc Intersections

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 $O(n^{4\mathrm{/}3+\epsilon })$ 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 $O(n^{4\mathrm{/}3}{\log }^{16\mathrm{/}3}n)$ time and improve it to $O(n^{1+\epsilon }+K^{1\mathrm{/}3}{n}^{2\mathrm{/}3}(\frac{n^2}{n+K}{)}^{\epsilon }{\log }^{16\mathrm{/}3}n)$ time for small $K$, where $K$ is the number of intersections of all arcs.