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Conflict-free Chromatic Art Gallery Coverage

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Abstract

We consider a chromatic variant of the art gallery problem, where each guard is assigned one of k distinct colors. A placement of such colored guards 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. What is the smallest number k(n) of colors that ensure a conflict-free covering of all n-vertex polygons? We call this the conflict-free chromatic art gallery problem. The problem is motivated by applications in distributed robotics and wireless sensor networks where colors indicate the wireless frequencies assigned to a set of covering "landmarks" in the environment so that a mobile robot can always communicate with at least one landmark in its line-of-sight range without interference. Our main result shows that k(n) is O(log n) for orthogonal and for monotone polygons, and O(log^2 n) for arbitrary simple polygons. By contrast, if all guards visible from each point must have distinct colors, then k(n)is Omega(n) for arbitrary simple polygons and Omega(sqrt(n)) for orthogonal polygons, as shown by Erickson and LaValle [Proc. of RSS 2011].

BibTeX - Entry

@InProceedings{brtschi_et_al:LIPIcs:2012:3395,
  author =	{Andreas  B{\"a}rtschi and Subhash Suri},
  title =	{{Conflict-free Chromatic Art Gallery Coverage}},
  booktitle =	{29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
  pages =	{160--171},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-35-4},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{14},
  editor =	{Christoph D{\"u}rr and Thomas Wilke},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2012/3395},
  URN =		{urn:nbn:de:0030-drops-33952},
  doi =		{http://dx.doi.org/10.4230/LIPIcs.STACS.2012.160},
  annote =	{Keywords: art gallery problem, conflict-free coloring, visibility}
}

Keywords: art gallery problem, conflict-free coloring, visibility
Seminar: 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)
Issue date: 2012
Date of publication: 2012


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