An Optimal Algorithm for Computing Many Faces in Line Arrangements

TL;DR

This paper presents an O(m^{2/3}n^{2/3}+(n+m)\log n) time algorithm that matches the lower bound under the algebraic decision tree model and thus is optimal.

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 $Ω(n^{4/3})$ 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].

Figures (7)

• Figure 1: Illustrating $F_p(L)$ (the grey cell), $\mathcal{U}(L_+(p))$ (blue segments), and the lower envelope of $L_-(p)$ (red segments).
• Figure 2: Illustrating the proof of the crucial observation in Lemma \ref{['lem:30']}. The three solid lines are those in $Q^*$. All blue points are those in $\mathcal{D}(\widehat{L}_{\sigma})$ while all red points are those in $\mathcal{D}(L_+(\sigma'))$.
• Figure 3: Illustrating the notation in the dual plane: $F^*_p(L)$, which is dual to $F_p(L)$, is composed of the blue edges between the two inner common tangents (the dashed segments).
• Figure 4: Illustrating the lower envelope $\mathcal{L}(\mathcal{H}_+(p^*))$ (the thick blue edges) of the convex hulls of $\mathcal{H}_+(p^*)$. The dashed segments inside convex hulls are their representative segments.
• Figure 5: Illustrating the two rays $\rho_1$ and $\rho_2$.

Theorems & Definitions (13)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Definition 1
• Lemma 6
• Lemma 7
• Definition 2
• Lemma 8