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Improved bounds for the conflict-free chromatic art gallery problem

Published: 08 June 2014 Publication History

Abstract

In chromatic variants of the art gallery problem, simple polygons are guarded with point guards that are assigned one of k colors each. We say these guards cover the polygon. Here we consider the conflict-free chromatic art gallery problem, first studied by Bärtschi and Suri (Algorithmica 2013): A covering of the polygon is conflict-free if each point of the polygon is seen by some guard whose color appears exactly once among the guards visible to that point. We are interested in the smallest number k(n) of colors that ensure such a covering for every n-vertex polygon.
It is known that k(n) is O(log n) for orthogonal and for monotone polygons, and O(log2 n) for arbitrary simple polygons. Our main contribution in this paper is an improvement of the upper bound on k(n) to O(log n) for simple polygons.
The bound is achieved through a partitioning of the polygon into weak visibility subpolygons, which is known as a window partition. In a weak visibility polygon, there is a boundary edge e such that each point of the polygon is seen by some point on e. We show for the first time for this special class of polygons an upper bound of O(log n). For the subpolygons of the window partition we prove a novel concept of independence that allows to reuse colors in independent subpolygons. Combining these results leads to the upper bound of O(log n) for arbitrary simple polygons.

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SOCG'14: Proceedings of the thirtieth annual symposium on Computational geometry
June 2014
588 pages
ISBN:9781450325943
DOI:10.1145/2582112
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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Published: 08 June 2014

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Author Tags

  1. art gallery problem
  2. conflict-free coloring
  3. polygon
  4. visibility

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SOCG'14 Paper Acceptance Rate 60 of 175 submissions, 34%;
Overall Acceptance Rate 625 of 1,685 submissions, 37%

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  • (2024)On colourability of polygon visibility graphsEuropean Journal of Combinatorics10.1016/j.ejc.2023.103820117(103820)Online publication date: Mar-2024
  • (2023)The Dispersive Art Gallery ProblemComputational Geometry10.1016/j.comgeo.2023.102054(102054)Online publication date: Oct-2023
  • (2019)On Conflict-Free Chromatic Guarding of Simple PolygonsCombinatorial Optimization and Applications10.1007/978-3-030-36412-0_49(601-612)Online publication date: 23-Nov-2019
  • (2019)Efficient Guarding of Polygons and TerrainsFrontiers in Algorithmics10.1007/978-3-030-18126-0_3(26-37)Online publication date: 9-Apr-2019
  • (2015)On Conflict-Free Multi-coloringAlgorithms and Data Structures10.1007/978-3-319-21840-3_9(103-114)Online publication date: 28-Jul-2015

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