An Optimal Algorithm for Computing Many Faces in Line Arrangements

TL;DR

This paper introduces an optimal algorithm for identifying faces in line arrangements containing specific points, matching theoretical lower bounds and improving computational efficiency for a classic geometric problem.

Contribution

It presents the first optimal algorithm for computing faces in line arrangements with points, matching the lower bound and improving previous methods.

Findings

Algorithm runs in O(m^{2/3}n^{2/3}+(n+m)log n) time.

Algorithm is proven to be optimal under the algebraic decision tree model.

When m=n, runtime is O(n^{4/3}), matching the combinatorial complexity.

Abstract

Given a set of $m$ points and a set of $n$ lines in the plane, we consider the problem of computing the faces of the arrangement of the lines that contain at least one point. In this paper, we present an $O(m^{2/3}n^{2/3}+(n+m)\log n)$ time algorithm for the problem. We also show that this matches the lower bound under the algebraic decision tree model and thus our algorithm is optimal. In particular, when $m=n$, the runtime is $O(n^{4/3})$, which matches the worst case combinatorial complexity $\Omega(n^{4/3})$ of all output faces. This is the first optimal algorithm since the problem was first studied more than three decades ago [Edelsbrunner, Guibas, and Sharir, SoCG 1988].