Counting Unit Circular Arc Intersections

TL;DR

This work resolves the long-standing problem of counting intersections among $n$ unit circular arcs of the same radius by introducing a hierarchical $(1/r)$-cutting framework combined with new geometric observations tailored to unit arcs. The algorithm decomposes the plane into cells, reduces per-cell counting to restricted bichromatic subproblems, and handles multiple intersection types with specialized subroutines, achieving a worst-case time of $O(n^{4/3}\log^{16/3}n)$ and an adaptive bound $O(n^{1+\epsilon}+K^{1/3}n^{2/3}(\frac{n^{2}}{n+K})^{\epsilon}\log^{16/3}n)$ for small $K$. It also extends to the bichromatic setting with the same complexities and discusses connections to the segment-intersection problem, where modern improvements via related cutting schemes apply. The paper advances the $n^{4/3}$ barrier with polylog refinements and introduces techniques (cuttings, pseudo-trapezoidal decompositions, and lune-based counting) that may apply to other unit-arc and arrangement-type problems in computational geometry.

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 $O(n^{4/3+ε})$ time [Agarwal, Pellegrini, and Sharir, SIAM J. Comput., 1993], for any constant $ε>0$. 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 $O(n^{1+ε}+K^{1/3}n^{2/3}(\frac{n^2}{n+K})^ε\log^{16/3}n)$ time for small $K$, where $K$ is the number of intersections of all arcs.

Figures (10)

• Figure 1: Illustrating a pseudo-trapezoid. The top (resp., bottom) side is a circular arc that is part of an input arc. The left (resp., right) side a vertical segment.
• Figure 2: The set $N(C)$ is a subset of the grey cells. Because the side-length of each cell is $1/\sqrt{2}$, the length between any point in $C$ and any point not in a grey cell must be larger than $1$.
• Figure 3: Illustrating Observation \ref{['obser:20']}: $p$ is the center of $s_r$. The two dashed circles are unit circles centered at the two endpoints of $s_b$, respectively. The grey area is $lune(s_b)$.
• Figure 4: Illustrating $s_r$, $s_r'$, and $z(s_r)$ for the case where $C_r$ is below $C$. The point $p$ is the center of $s_r$ and the dashed curve is the upper semicircle of $\alpha(s_r)$. In this figure, $s_r$ and $s_b$ form a type (3.1.2.1) intersection (i.e., $s_r$ and $s_b$ intersect exactly once and $s_r'$ does not intersect $s_b$).
• Figure 5: Illustrating $s_r$, $s_r'$, and $z(s_r)$ for the case where $C_r$ is below $C$. The point $p$ is the center of $s_r$ and the dashed curve is the upper semicircle of $\alpha(s_r)$. In this figure, $s_r$ and $s_b$ form a type (3.1.2.2) intersection (i.e., $s_b$ and $s_r$ intersect exactly once and $s_b$ intersects $s_r'$).

Theorems & Definitions (12)

• Theorem 1
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9