Watchman Routes for Lines and Segments

  • Adrian Dumitrescu
  • Joseph S. B. Mitchell
  • Paweł Żyliński
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7357)

Abstract

Given a set \(\mathcal L\) of non-parallel lines, a watchman route (tour) for \(\mathcal L\) is a closed curve contained in the union of the lines in \(\mathcal L\) such that every point on any line is visible (along a line) from at least one point of the route; similarly, we define a watchman route (tour) for a connected set \(\mathcal S\) of line segments. The watchman route problem for a given set of lines or line segments is to find a shortest watchman route for the input set, and these problems are natural special cases of the watchman route problem in multiply connected polygonal domains.

In this paper, we show that the problem of computing a shortest watchman route for a set of n non-parallel lines in the plane is polynomially tractable, while it becomes NP-hard in 3D. Then, we reprove NP-hardness of this problem for line segments in the plane and provide a polynomial-time approximation algorithm with ratio O(log3n). Additionally, we consider some special cases of the watchman route problem on line segments, for which we provide improved approximation or exact algorithms.

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References

  1. 1.
    Brimkov, V.E., Leach, A., Mastroianni, M., Wu, J.: Guarding a set of line segments in the plane. Theoretical Computer Science 412(15), 1313–1324 (2011)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Chin, W., Ntafos, S.: Optimum watchman routes. In: Proc. 2nd Symposium on Computational Geometry, pp. 24–33 (1986)Google Scholar
  3. 3.
    Chin, W., Ntafos, S.: Optimum watchman routes. Information Processing Letters 28(1), 39–44 (1988)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Chvátal, V.: A combinatorial theorem in plane geometry. Journal of Combinatorial Theory, Series B 18, 39–41 (1997)CrossRefGoogle Scholar
  5. 5.
    Dror, M., Efrat, A., Lubiw, A., Mitchell, J.S.B.: Touring a sequence of polygons. In: Proc. 35th Symposium on Theory of Computing, pp. 473–482 (2003)Google Scholar
  6. 6.
    Dumitrescu, A., Mitchell, J.S.B.: Approximation algorithms for TSP with neighborhoods in the plane. Journal of Algorithms 48(1), 135–159 (2003)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Dumitrescu, A., Tóth, C.D.: Watchman tours for polygons with holes. Computational Geometry: Theory and Applications 45(7), 326–333 (2012)MATHCrossRefGoogle Scholar
  8. 8.
    Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. In: Proc. 35th ACM Symposium on Theory of Computing, pp. 448–455 (2003)Google Scholar
  9. 9.
    Garey, M.R., Graham, R., Johnson, D.S.: Some NP-complete geometric problems. In: Proc. 8th ACM Symposium on Theory of Computing, pp. 10–22 (1976)Google Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Co., New York (1979)MATHGoogle Scholar
  11. 11.
    Gewali, L.P., Ntafos, S.: Covering grids and orthogonal polygons with periscope guards. Computational Geometry: Theory and Applications 2(6), 309–334 (1993)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Hoffmann, F.: Private communication. In: The European Workshop on Computational Geometry (EuroCG 2000), Eilat, Israel (2000)Google Scholar
  13. 13.
    Kosowski, A., Małafiejski, M., Żyliński, P.: Cooperative mobile guards in grids. Computational Geometry: Theory and Applications 37(2), 59–71 (2007)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Mata, C.S., Mitchell, J.S.B.: Approximation algorithms for geometric tour and network design problems. In: Proc. 11th Symposium on Computational Geometry, pp. 360–369 (1995)Google Scholar
  15. 15.
    Mitchell, J.S.B.: Geometric shortest paths and network optimization. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 633–701. Elsevier (2000)Google Scholar
  16. 16.
    Ntafos, S.: On gallery watchmen in grids. Information Processing Letters 23, 99–102 (1986)MATHCrossRefGoogle Scholar
  17. 17.
    O’Rourke, J.: Art Gallery Theorems and Algorithms. Oxford University Press, New York (1987)MATHGoogle Scholar
  18. 18.
    Papadimitriou, C.H.: Euclidean TSP is NP-complete. Theoretical Computer Science 4, 237–244 (1977)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Shermer, T.: Recent results in Art Galleries. Proc. of the IEEE 80, 1384–1399 (1992)CrossRefGoogle Scholar
  20. 20.
    Slavik, P.: The errand scheduling problem, CSE Technical Report 97-02, University of Buffalo (1997)Google Scholar
  21. 21.
    Tan, X.: Fast computation of shortest watchman routes in simple polygons. Information Processing Letters 77(1), 27–33 (2001)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Tan, X.: A linear-time 2-approximation algorithm for the watchman route problem for simple polygons. Theoretical Computer Science 384(1), 92–103 (2007)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Tóth, C.D.: Illumination in the presence of opaque line segments in the plane. Computational Geometry: Theory and Applications 21(3), 193–204 (2002)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Tóth, C.D.: Illuminating disjoint line segments in the plane. Discrete & Computational Geometry 30(3), 489–505 (2003)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Urrutia, J.: Art gallery and illumination problems. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 973–1027. Elsevier (2000)Google Scholar
  26. 26.
    Xu, N., Brass, P.: On the complexity of guarding problems on orthogonal arrangements. In: Abstracts of the 20th Fall Workshop on Computational Geometry (FWCG 2010), #33 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Adrian Dumitrescu
    • 1
  • Joseph S. B. Mitchell
    • 2
  • Paweł Żyliński
    • 3
  1. 1.Dept. of Computer ScienceUniv. of Wisconsin–MilwaukeeUSA
  2. 2.Dept. of Appl. Math. and StatisticsState Univ. of NY at Stony BrookUSA
  3. 3.Inst. of InformaticsUniversity of GdańskPoland

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