An Optimal Algorithm for Computing Many Faces in Line Arrangements

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].