Algorithmica

, Volume 57, Issue 3, pp 499–516 | Cite as

Improved Bounds for Wireless Localization

  • Tobias Christ
  • Michael Hoffmann
  • Yoshio Okamoto
  • Takeaki Uno
Article
  • 104 Downloads

Abstract

We consider a novel class of art gallery problems inspired by wireless localization that has recently been introduced by Eppstein, Goodrich, and Sitchinava. Given a simple polygon P, place and orient guards each of which broadcasts a unique key within a fixed angular range. In contrast to the classical art gallery setting, broadcasts are not blocked by the edges of P. At any point in the plane one must be able to tell whether or not one is located inside P only by looking at the set of keys received. In other words, 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 problem where guards may be placed anywhere by showing that the maximum number of guards to describe any simple polygon on n vertices is between roughly \(\frac{3}{5}n\) and \(\frac{4}{5}n\) . A guarding that uses at most \(\frac{4}{5}n\) guards can be obtained in O(nlog n) time. 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.

Keywords

Computational geometry Art gallery problems 

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References

  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. Comb. Theory Ser. B 18, 39–41 (1975) MATHCrossRefGoogle 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. http://arxiv.org/pdf/0709.3554v1 (2007)
  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) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Eppstein, D., Goodrich, M.T., Sitchinava, N.: Guard placement for efficient point-in-polygon proofs. In: Proc. of the 23rd Annu. Sympos. Comput. Geom., pp. 27–36 (2007) Google Scholar
  6. 6.
    Fisk, S.: A short proof of Chvátal’s watchman theorem. J. Comb. Theory Ser. B 24, 374 (1978) MATHCrossRefMathSciNetGoogle 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) MATHMathSciNetGoogle Scholar
  8. 8.
    Guibas, L.J., Hershberger, J., Snoeyink, J.: Compact interval trees: a data structure for convex hulls. Int. J. Comput. Geom. Appl. 1(1), 1–22 (1991) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kahn, J., Klawe, M.M., Kleitman, D.J.: Traditional galleries require fewer watchmen. SIAM J. Algebr. Discrete Methods 4, 194–206 (1983) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Karavelas, M.I.: Guarding curvilinear art galleries with edge or mobile guards. In: Proc. of the 13th ACM Solid and Physical Modeling Symposium, pp. 339–345 (2008) Google Scholar
  11. 11.
    Lee, D.T., Lin, A.K.: Computational complexity of art gallery problems. IEEE Trans. Inf. Theory 32(2), 276–282 (1986) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    O’Rourke, J.: Galleries need fewer mobile guards: a variation on Chvátal’s theorem. Geom. Dedicata 14, 273–283 (1983) MATHMathSciNetGoogle Scholar
  13. 13.
    O’Rourke, J.: Visibility. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, Chap. 28, pp. 643–663. CRC, Boca Raton (2004) Google Scholar
  14. 14.
    O’Rourke, J.: Computational geometry column 48. ACM SIGACT News 37(3), 55–57 (2006) CrossRefGoogle Scholar
  15. 15.
    Shermer, T.C.: Recent results in art galleries. Proc. IEEE 80(9), 1384–1399 (1992) CrossRefGoogle Scholar
  16. 16.
    Speckmann, B., Tóth, C.D.: Allocating vertex pi-guards in simple polygons via pseudo-triangulations. Discrete Comput. Geom. 33(2), 345–364 (2005) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Urrutia, J.: Art gallery and illumination problems. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 973–1027. Elsevier Science/North-Holland, Amsterdam (2000) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

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

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