, Volume 5, Issue 1, pp 427-448

Construction of ɛ-nets

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Abstract

LetS be a set ofn points in the plane and letɛ be a real number, 0<ɛ<1. We give a deterministic algorithm, which in timeO( −2 log(1/ɛ)+ɛ −8) (resp.O( −2 log(1/ɛ)+ɛ −10) constructs anɛ-netNS of sizeO((1/ɛ) (log(1/ɛ))2) for intersections ofS with double wedges (resp. triangles); this means that any double wedge (resp. triangle) containing more thatɛn points ofS contains a point ofN. This givesO(n logn) deterministic preprocessing for the simplex range-counting algorithm of Haussler and Welzl [HW] (in the plane).

We also prove that given a setL ofn lines in the plane, we can cut the plane intoO(ɛ −2) triangles in such a way that no triangle is intersected by more thanɛn lines ofL. We give a deterministic algorithm for this with running timeO( −2 log(1/ɛ)). This has numerous applications in various computational geometry problems.