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Israel Journal of Mathematics

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

Guarding galleries where no point sees a small area

  • Pavel Valtr
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.

Keywords

Lebesgue Measure Random Point Computational Geometry Range Query System Versus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Hebrew University 1998

Authors and Affiliations

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

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