Israel Journal of Mathematics

, Volume 104, Issue 1, pp 1–16 | Cite as

Guarding galleries where no point sees a small area

Article

Abstract

We prove the following result which is the planar version of a conjecture of Kavraki, Latombe, Motwani, and Raghavan: there is a functionf(h, ɛ) polynomial inh and 1/ɛ such that ifX is a compact planar set of Lebesgue measure 1 withh holes, such that any pointxX sees a part ofX of measure at leastɛ, then there is a setG of at mostf(h, ɛ) points (guards) inX such that any point ofX is seen by at least one point ofG. With a high probability, a setG off(h, ɛ) random points inX (chosen uniformly and independently) has the above property.

In the proof (givingf(h, ɛ)≤(2+o(1))1/ɛ log 1/ɛ log2 h) we apply ideas of Kalai and Matoušek who proved a weaker boundf(h, ɛ)≤C(h)1/ɛ log 1/ɛ, whereC(h) is a ‘quite fast growing function’ ofh. We improve their bound by showing a stronger result on the so-called VC-dimension of related set systems.

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References

  1. [Ed]
    H. Edelsbrunner,Computational Geometry, Springer-Verlag, Berlin, 1987.Google Scholar
  2. [HW]
    D. Haussler and E. Welzl,Epsilon-nets and simplex range queries, Discrete and Computational Geometry2 (1987), 127–151.CrossRefMathSciNetMATHGoogle Scholar
  3. [KM]
    G. Kalai and J. Matoušek,Guarding galleries where every point sees a large area, Israel Journal of Mathematics101 (1997), 125–139.CrossRefMathSciNetMATHGoogle Scholar
  4. [KLMR]
    L. Kavraki, J-C. Latombe, R. Motwani and P. Raghavan,Randomized query processing in robot motion planning, Proceedings of the 27th ACM Symposium on Theoretical Computation, 1995.Google Scholar
  5. [KPW]
    J. Komlós, J. Pach and G. Woeginger,Almost tight bounds for ɛ-nets, Discrete and Computational Geometry7 (1992), 163–173.CrossRefMathSciNetMATHGoogle Scholar
  6. [V]
    P. Valtr,On galleries with no bad points, submitted.Google Scholar
  7. [VC]
    V. N. Vapnik and A. Ya. Chervonenkis,On the uniform convergence of relative frequencies of events to their probabilities, Theory of Probability and its Applications16 (1971), 264–280.CrossRefMATHGoogle Scholar

Copyright information

© Hebrew University 1998

Authors and Affiliations

  1. 1.Department of Applied MathematicsCharles UniversityPraha 1Czech Republic

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