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Algorithmica

, Volume 81, Issue 1, pp 69–97 | Cite as

Guarding Orthogonal Art Galleries with Sliding k-Transmitters: Hardness and Approximation

  • Therese Biedl
  • Timothy M. Chan
  • Stephanie Lee
  • Saeed MehrabiEmail author
  • Fabrizio Montecchiani
  • Hamideh Vosoughpour
  • Ziting Yu
Article
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Abstract

A sliding k-transmitter inside an orthogonal polygon P, for a fixed \(k\ge 0\), is a point guard that travels along an axis-parallel line segment s in P. The sliding k-transmitter can see a point \(p\in P\) if the perpendicular from p onto s intersects the boundary of P in at most k points. In the Minimum Sliding k-Transmitters (\({\hbox {ST}}_k\)) problem, the objective is to guard P with the minimum number of sliding k-transmitters. In this paper, we give a constant-factor approximation algorithm for the \({\hbox {ST}}_k\) problem on P for any fixed \(k\ge 0\). Moreover, we show that the \({\hbox {ST}}_0\) problem is NP-hard on orthogonal polygons with holes even if only horizontal sliding 0-transmitters are allowed. For \(k>0\), the problem is NP-hard even in the extremely restricted case where P is simple and monotone. Finally, we study art gallery theorems; i.e., we give upper and lower bounds on the number of sliding transmitters required to guard P relative to the number of vertices of P.

Keywords

Art gallery problem Sliding k-transmitters \(\epsilon \)-nets 
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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Therese Biedl
    • 1
  • Timothy M. Chan
    • 1
  • Stephanie Lee
    • 1
  • Saeed Mehrabi
    • 1
    Email author
  • Fabrizio Montecchiani
    • 2
  • Hamideh Vosoughpour
    • 1
  • Ziting Yu
    • 1
  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Department of EngineeringUniversity of PerugiaPerugiaItaly

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