The complexity of the outer face in arrangements of random segments

Noga Alon*, Dan Halperin, Oren Nechushtan, Micha Sharir

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

We investigate the complexity of the outer face in arrangements of line segments of a fixed length ℓ in the plane, drawn uniformly at random within a square. We derive upper bounds on the expected complexity of the outer face, and establish a certain phase transition phenomenon during which the expected complexity of the outer face drops sharply as a function of the total number of segments. In particular we show that up till the phase transition the complexity of the outer face is almost linear in n, and that after the phase transition, the complexity of the outer face is roughly proportional to √n. Our study is motivated by the analysis of a practical point-location algorithm (so-called walk-along-a-line point-location algorithm) and indeed, it explains experimental observations of the behavior of the algorithm on arrangements of random segments.

Original languageEnglish
Title of host publicationProceedings of the 24th Annual Symposium on Computational Geometry 2008, SCG'08
Pages69-78
Number of pages10
DOIs
StatePublished - 2008
Event24th Annual Symposium on Computational Geometry, SCG'08 - College Park, MD, United States
Duration: 9 Jun 200811 Jun 2008

Publication series

NameProceedings of the Annual Symposium on Computational Geometry

Conference

Conference24th Annual Symposium on Computational Geometry, SCG'08
Country/TerritoryUnited States
CityCollege Park, MD
Period9/06/0811/06/08

Keywords

  • Algorithms
  • Theory

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