Improved Bounds for Wireless Localization

  • Tobias Christ
  • Michael Hoffmann
  • Yoshio Okamoto
  • Takeaki Uno
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5124)


We consider a novel class of art gallery problems inspired by wireless localization. Given a simple polygon P, place and orient guards each of which broadcasts a unique key within a fixed angular range. Broadcasts are not blocked by the edges of P. The interior of the polygon must be described by a monotone Boolean formula composed from the keys. We improve both upper and lower bounds for the general setting by showing that the maximum number of guards to describe any simple polygon on n vertices is between roughly \({\frac{3}{5}}\)n and \(\frac{4}{5}\)n. For the natural setting where guards may be placed aligned to one edge or two consecutive edges of P only, we prove that n − 2 guards are always sufficient and sometimes necessary.


Convex Hull Simple Polygon Extremal Vertex Polygonal Chain Improve Bound 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aigner, M., Ziegler, G.M.: Proofs from THE BOOK, 3rd edn. Springer, Berlin (2003)Google Scholar
  2. 2.
    Chvátal, V.: A Combinatorial Theorem in Plane Geometry. J. Combin. Theory Ser. B 18, 39–41 (1975)CrossRefzbMATHGoogle Scholar
  3. 3.
    Damian, M., Flatland, R., O’Rourke, J., Ramaswami, S.: A New Lower Bound on Guard Placement for Wireless Localization. In: 17th Annual Fall Workshop on Computational Geometry (2007)Google Scholar
  4. 4.
    Dobkin, D.P., Guibas, L., Hershberger, J., Snoeyink, J.: An Efficient Algorithm for Finding the CSG Representation of a Simple Polygon. Algorithmica 10, 1–23 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Eppstein, D., Goodrich, M.T., Sitchinava, N.: Guard Placement for Efficient Point-in-Polygon Proofs. In: Proc. 23rd Annu. Sympos. Comput. Geom. pp. 27–36 (2007)Google Scholar
  6. 6.
    Fisk, S.: A Short Proof of Chvátal’s Watchman Theorem. J. Combin. Theory Ser. B 24, 374 (1978)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Fragoudakis, C., Markou, E., Zachos, S.: Maximizing the Guarded Boundary of an Art Gallery is APX-complete. Comput. Geom. Theory Appl. 38(3), 170–180 (2007)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Kahn, J., Klawe, M.M., Kleitman, D.J.: Traditional Galleries Require Fewer Watchmen. SIAM J. Algebraic Discrete Methods 4, 194–206 (1983)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Karavelas, M.I., Tsigaridas, E.P.: Guarding Curvilinear Art Galleries with Vertex or Point Guards. Rapport de recherche 6132, INRIA (2007)Google Scholar
  10. 10.
    Lee, D.T., Lin, A.K.: Computational Complexity of Art Gallery Problems. IEEE Trans. Inform. Theory 32(2), 276–282 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    O’Rourke, J.: Galleries Need Fewer Mobile Guards: A Variation on Chvátal’s Theorem. Geom. Dedicata 14, 273–283 (1983)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    O’Rourke, J.: Visibility. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, ch. 28, pp. 643–663. CRC Press LLC, Boca Raton (2004)Google Scholar
  13. 13.
    O’Rourke, J.: Computational Geometry Column 48. ACM SIGACT News 37(3), 55–57 (2006)CrossRefGoogle Scholar
  14. 14.
    Shermer, T.C.: Recent Results in Art Galleries. Proc. IEEE 80(9), 1384–1399 (1992)CrossRefGoogle Scholar
  15. 15.
    Speckmann, B., Tóth, C.D.: Allocating Vertex Pi-guards in Simple Polygons via Pseudo-triangulations. Discrete Comput. Geom. 33(2), 345–364 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Urrutia, J.: Art Gallery and Illumination Problems. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 973–1027. Elsevier Science Publishers B.V, Amsterdam (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Tobias Christ
    • 1
  • Michael Hoffmann
    • 1
  • Yoshio Okamoto
    • 2
  • Takeaki Uno
    • 3
  1. 1.Institute for Theoretical Computer ScienceETH ZürichSwitzerland
  2. 2.Tokyo Institute of TechnologyJapan
  3. 3.National Institute of InformaticsTokyoJapan

Personalised recommendations