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 pointx ∈X 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
H. Edelsbrunner,Computational Geometry, Springer-Verlag, Berlin, 1987.
D. Haussler and E. Welzl,Epsilon-nets and simplex range queries, Discrete and Computational Geometry2 (1987), 127–151.
G. Kalai and J. Matoušek,Guarding galleries where every point sees a large area, Israel Journal of Mathematics101 (1997), 125–139.
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.
J. Komlós, J. Pach and G. Woeginger,Almost tight bounds for ɛ-nets, Discrete and Computational Geometry7 (1992), 163–173.
P. Valtr,On galleries with no bad points, submitted.
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.
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This research was supported by Czech Republic Grant 0194 and by Charles University grants No. 193 and 194.
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Valtr, P. Guarding galleries where no point sees a small area. Isr. J. Math. 104, 1–16 (1998). https://doi.org/10.1007/BF02897056
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DOI: https://doi.org/10.1007/BF02897056