Constructing Many Faces in Arrangements of Lines and Segments

Document Type

Conference Paper

Journal/Book Title/Conference

2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

Publisher

Society for Industrial and Applied Mathematics

Publication Date

2022

Award Number

NSF, Division of Computing and Communication Foundations (CCF) 2005323

Funder

NSF, Division of Computing and Communication Foundations (CCF)

First Page

3168

Last Page

3180

Abstract

We present new algorithms for computing many faces in arrangements of lines and segments. Given a set S of n lines (resp., segments) and a set P of m points in the plane, the problem is to compute the faces of the arrangements of S that contain at least one point of P. For the line case, we give a deterministic algorithm of time. This improves the previously best deterministic algorithm [Agarwal, 1990] by a factor of log2.22 n and improves the previously best randomized algorithm [Agarwal, Matoušek, and Schwarzkopf, 1998] by a factor of log1/3 n in certain cases (e.g., when m = Θ(n)). For the segment case, we present a deterministic algorithm of time, where and α(n) is the inverse Ackermann function. This improves the previously best deterministic algorithm [Agarwal, 1990] by a factor of log2.11 n and improves the previously best randomized algorithm [Agarwal, Matoušek, and Schwarzkopf, 1998] by a factor of log n in certain cases (e.g., when m = Θ(n)). We also give a randomized algorithm of O(m2/3 K1/3 log n + τ(nα(n) + n log m + m) log n log K) expected time, where K is the number of intersections of all segments of S. In addition, we consider the query version of the problem, that is, preprocess S to compute the face of the arrangement of S that contains any query point. We present new results that improve the previous work for both the line and the segment cases.

Link to Full Text

Share

COinS