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Guard placement for efficient point-in-polygon proofs

Published: 06 June 2007 Publication History

Abstract

We consider the problem of placing a small number of angle guards inside a simple polygon P so asto provide efficient proofs that any given point is inside P. Each angle guard views an infinite wedge of the plane, and a point can prove membership in P if it is inside the wedges for a set of guards whose common intersection contains no points outside the polygon. This model leads to a broad class of new art gallery type problems, which we call "sculpture garden" problems and for which we provide upper and lower bounds. In particular, we show there is a polygon P such that a "natural" angle-guard vertex placement cannot fully distinguish between pointson the inside and outside of P (even if we place a guard at every vertex of P), which implies that Steiner-point guards are sometimes necessary. More generally, we show that, for any polygon P, there is a set of n+2(h-1) angle guards that solve the sculpture garden problem for P, where h is the number of holes in P (so a simple polygon can be defined with n-2 guards). In addition, we show that, for any orthogonal polygon P, the sculpture garden problem can besolved using n/2 angle guards. We also give an example of a class of simple (non-general-position) polygons that have sculpture garden solutions using O(√n) guards, and we show this bound is optimal to within a constant factor. Finally, while optimizing the number of guards solving a sculpture garden problem for a particular P is of unknown complexity, we show how to find in polynomial time a guard placement whose size is within a factor of 2 of the optimal number for any particular polygon.

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  • (2017)Time-Space Trade-Off for Finding the k-Visibility Region of a Point in a PolygonWALCOM: Algorithms and Computation10.1007/978-3-319-53925-6_24(308-319)Online publication date: 21-Feb-2017
  • (2011)Guarding curvilinear art galleries with edge or mobile guards via 2-dominance of triangulation graphsComputational Geometry: Theory and Applications10.1016/j.comgeo.2010.07.00244:1(20-51)Online publication date: 1-Jan-2011
  • (2010)Improved Bounds for Wireless LocalizationAlgorithmica10.5555/3118227.311847857:3(499-516)Online publication date: 1-Jul-2010
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cover image ACM Conferences
SCG '07: Proceedings of the twenty-third annual symposium on Computational geometry
June 2007
404 pages
ISBN:9781595937056
DOI:10.1145/1247069
  • Program Chair:
  • Jeff Erickson
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 ACM 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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Publication History

Published: 06 June 2007

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Author Tags

  1. angle guards
  2. approximation algorithm
  3. art gallery problem
  4. floodlights
  5. localization

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Cited By

View all
  • (2017)Time-Space Trade-Off for Finding the k-Visibility Region of a Point in a PolygonWALCOM: Algorithms and Computation10.1007/978-3-319-53925-6_24(308-319)Online publication date: 21-Feb-2017
  • (2011)Guarding curvilinear art galleries with edge or mobile guards via 2-dominance of triangulation graphsComputational Geometry: Theory and Applications10.1016/j.comgeo.2010.07.00244:1(20-51)Online publication date: 1-Jan-2011
  • (2010)Improved Bounds for Wireless LocalizationAlgorithmica10.5555/3118227.311847857:3(499-516)Online publication date: 1-Jul-2010
  • (2010)Coverage with k-transmitters in the presence of obstaclesProceedings of the 4th international conference on Combinatorial optimization and applications - Volume Part II10.5555/1940424.1940425(1-15)Online publication date: 18-Dec-2010
  • (2010)Coverage with k-Transmitters in the Presence of ObstaclesCombinatorial Optimization and Applications10.1007/978-3-642-17461-2_1(1-15)Online publication date: 2010
  • (2009)Guarding curvilinear art galleries with vertex or point guardsComputational Geometry: Theory and Applications10.1016/j.comgeo.2008.11.00242:6-7(522-535)Online publication date: 1-Aug-2009
  • (2009)Improved Bounds for Wireless LocalizationAlgorithmica10.1007/s00453-009-9287-257:3(499-516)Online publication date: 14-Feb-2009
  • (2008)Guarding curvilinear art galleries with edge or mobile guardsProceedings of the 2008 ACM symposium on Solid and physical modeling10.1145/1364901.1364950(339-345)Online publication date: 2-Jun-2008
  • (2008)Improved Bounds for Wireless LocalizationProceedings of the 11th Scandinavian workshop on Algorithm Theory10.1007/978-3-540-69903-3_9(77-89)Online publication date: 2-Jul-2008

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