Problems of posting sentries: Variations on the art gallery theorem

Detailed abstract
  • R. P. Krishnaswamy
  • C. E. Kim
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 318)


We examine the following problem and its variations: “Given a collection of mutually disjoint polygons (the objects) properly contained in a polygon (the enclosure), what is the minimum number of stationary guards that need to be posted within the enclosure but outside the objects so that every edge of each object is seen by some guard”.

Algorithms to compute minimal postings for a convex object in a convex enclosure and a convex object in a nonconvex enclosure are presented. We prove that the problems of computing minimal postings for both a collection of convex objects in a convex enclosure, and a single nonconvex object in a convex enclosure, are NP-Hard.

The variation of posting a given number of guards for a convex object in a convex enclosure so as to minimize a measure of proximity of guards to points of the object is considered. Results for two different measures of proximity are presented.

Finally we show that the 3-dimensional problem of computing a posting with a minimum number of guards for a convex polyhedral object in a convex polyhedral enclosure is NP-Hard. Upperbounds on the number of guards required to cover every point on the surface of the object are derived.


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  1. [1]
    Alok Aggarwal ‘The Art Gallery Theorem and Algorithms’ Ph.D thesis, Johns Hopkins University, 1984.Google Scholar
  2. [2]
    B. Bollobas ‘Graph Theory’ Springer-Verlag, 1979.Google Scholar
  3. [3]
    V. Chvatal ‘A combinatorial theorem in plane geometry’ Journal of Combinatorial Theory B, B-18 pp. 39–41, 1975.Google Scholar
  4. [4]
    Norshinge Chiba, Kazuneri Onoguchi and Tako Nishizeki ‘Drawing plane graphs nicely’ Acta Informatica, Vol 22 pp. 187–201, 1985.Google Scholar
  5. [5]
    S. Fisk ‘A short proof of Chvatals watchman theorem’ Journal of combinatorial theory B, B-24 pp. 374, 1978.Google Scholar
  6. [6]
    M. Garey and D.S. Johnson ‘Computers and intractibility, a guide to the theory of NP-completeness’ W.H. Freeman and co., 1979.Google Scholar
  7. [7]
    R. Krishnaswamy ‘Problems of posting sentries’ Ph.D thesis, Washington State University, Dec. 1987.Google Scholar
  8. [8]
    J. Kahn, M. Klawe and D. Kleitman ‘Traditional galleries require fewer watchmen’ SIAM J. of Alg. and disc. meth., Vol 4 pp. 194–206, 1983.Google Scholar
  9. [9]
    D.T. Lee and A.K. Lin ‘Computational complexity of art gallery problems’ IEEE Trans. Info. Theory, Vol 32 pp. 276–282, 1986.Google Scholar
  10. [10]
    J. O'Rourke ‘An alternate proof of the rectilinear art gallery theorem’ Journal of geometry, Vol 21 pp. 118–130, 1983.Google Scholar
  11. [11]
    J. O'Rourke ‘Galleries need fewer mobile guards; a variation on Chvatals watchmen Theorem’ Journal of geometry, Vol 21 pp. 273–283, 1983.Google Scholar
  12. [12]
    F.P. Preperata and M. I. Shamos ‘Computational geometry’ Springer-Verlag, 1985.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • R. P. Krishnaswamy
    • 1
  • C. E. Kim
    • 2
  1. 1.Computer Science Dept.University of WyomingLaramie
  2. 2.Computer Science Dept.Washington State UniversityPullman

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