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Coverage with k-Transmitters in the Presence of Obstacles

  • Brad Ballinger
  • Nadia Benbernou
  • Prosenjit Bose
  • Mirela Damian
  • Erik D. Demaine
  • Vida Dujmović
  • Robin Flatland
  • Ferran Hurtado
  • John Iacono
  • Anna Lubiw
  • Pat Morin
  • Vera Sacristán
  • Diane Souvaine
  • Ryuhei Uehara
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6509)

Abstract

For a fixed integer k ≥ 0, a k-transmitter is an omnidirectional wireless transmitter with an infinite broadcast range that is able to penetrate up to k “walls”, represented as line segments in the plane. We develop lower and upper bounds for the number of k-transmitters that are necessary and sufficient to cover a given collection of line segments, polygonal chains and polygons.

Keywords

Convex Polygon Simple Polygon Visibility Graph Entire Plane Polygonal Region 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Aichholzer, O., Aurenhammer, F., Hurtado, F., Ramos, P., Urrutia, J.: k-convex polygons. In: EuroCG, pp. 117–120 (2009)Google Scholar
  2. 2.
    Aichholzer, O., Fabila-Monroy, R., Flores-Pealoza, D., Hackl, T., Huemer, C., Urrutia, J., Vogtenhuber, B.: Modem illumination of monotone polygons. In: EuroCG (2009)Google Scholar
  3. 3.
    Borodin, O.: A new proof of the 6 color theorem. Journal of Graph Theory 19(4), 507–521 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Christ, T., Hoffmann, M., Okamoto, Y., Uno, T.: Improved bounds for wireless localization. In: Gudmundsson, J. (ed.) SWAT 2008. LNCS, vol. 5124, pp. 77–89. Springer, Heidelberg (2008)Google Scholar
  5. 5.
    Chvátal, V.: A combinatorial theorem in plane geometry. Journal of Combinatorial Theory Series B 18, 39–41 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Czyzowicz, J., Rivera-Campo, E., Santoro, N., Urrutia, J., Zaks, J.: Guarding rectangular art galleries. Discrete Applied Math. 50, 149–157 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Damian, M., Flatland, R., O’Rourke, J., Ramaswami, S.: A new lower bound on guard placement for wireless localization. In: Proc. of the 17th Fall Workshop on Computational Geometry, FWCG 2007, pp. 21–24 (November 2007)Google Scholar
  8. 8.
    Dean, A.M., Evans, W., Gethner, E., Laison, J., Safari, M.A., Trotter, W.T.: Bar k-visibility graphs: Bounds on the number of edges, chromatic number, and thickness. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 73–82. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Eppstein, D., Goodrich, M.T., Sitchinava, N.: Guard placement for efficient point-in-polygon proofs. In: SoCG, pp. 27–36 (2007)Google Scholar
  10. 10.
    Fabila-Monroy, R., Vargas, A.R., Urrutia, J.: On modem illumination problems. In: XIII Encuentros de Geometria Computacional, Zaragoza, Spain (June 2009)Google Scholar
  11. 11.
    Felsner, S., Massow, M.: Parameters of bar k-visibility graphs. Journal of Graph Algorithms and Applications 12(1), 5–27 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fulek, R., Holmsen, A.F., Pach, J.: Intersecting convex sets by rays. Discrete Comput. Geom. 42(3), 343–358 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hartke, S.G., Vandenbussche, J., Wenger, P.: Further results on bar k-visibility graphs. SIAM Journal of Discrete Mathematics 21(2), 523–531 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lee, D.T., Lin, A.K.: Computational complexity of art gallery problems. IEEE Trans. Inf. Theor. 32(2), 276–282 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    O’Rourke, J.: Art gallery theorems and algorithms. Oxford University Press, Inc., New York (1987)zbMATHGoogle Scholar
  16. 16.
    Urrutia, J.: Art gallery and illumination problems. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 973–1027. North-Holland, Amsterdam (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Brad Ballinger
    • 1
  • Nadia Benbernou
    • 2
  • Prosenjit Bose
    • 3
  • Mirela Damian
    • 4
  • Erik D. Demaine
    • 2
  • Vida Dujmović
    • 3
  • Robin Flatland
    • 5
  • Ferran Hurtado
    • 6
  • John Iacono
    • 7
  • Anna Lubiw
    • 8
  • Pat Morin
    • 3
  • Vera Sacristán
    • 6
  • Diane Souvaine
    • 9
  • Ryuhei Uehara
    • 10
  1. 1.Humboldt State UniversityArcataUSA
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA
  3. 3.Carleton UniversityOttawaCanada
  4. 4.Villanova UniversityVillanovaUSA
  5. 5.Siena CollegeLoudonvilleUSA
  6. 6.Universitat Politècnica de CatalunyaBarcelonaSpain
  7. 7.Polytechnic Institute of New York UniversityNew YorkUSA
  8. 8.University of WaterlooWaterlooCanada
  9. 9.Tufts UniversityMedfordUSA
  10. 10.Japan Advanced Institute of Science and TechnologyIshikawaJapan

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