Discrete & Computational Geometry

, Volume 33, Issue 2, pp 345–364

Allocating Vertex π-Guards in Simple Polygons via Pseudo-Triangulations

Article

DOI: 10.1007/s00454-004-1091-9

Cite this article as:
Speckmann, B. & Tóth, C. Discrete Comput Geom (2005) 33: 345. doi:10.1007/s00454-004-1091-9

Abstract

We use the concept of pointed pseudo-triangulations to establish new upper and lower bounds on a well known problem from the area of art galleries: What is the worst case optimal number of vertex π-guards that collectively monitor a simple polygon with n vertices? Our results are as follows: (1) Any simple polygon with n vertices can be monitored by at most \lfloor n/2 \rfloor general vertex π-guards. This bound is tight up to an additive constant of 1. (2) Any simple polygon with n vertices, k of which are convex, can be monitored by at most \lfloor (2n – k)/3 \rfloor edge-aligned vertexπ-guards. This is the first non-trivial upper bound for this problem and it is tight for the worst case families of polygons known so far.

Copyright information

© Springer Science+Business Media, Inc. 2004

Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science, TU Eindhoven, P.O. Box 513, Eindhoven 5600 MBThe Netherlands
  2. 2.Department of Computer Science, University of California at Santa Barbara, Santa Barbara, CA 93106USA