Guarding galleries where no point sees a small area
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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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- [Ed]H. Edelsbrunner,Computational Geometry, Springer-Verlag, Berlin, 1987.Google Scholar
- [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
- [V]P. Valtr,On galleries with no bad points, submitted.Google Scholar