How to Net a Convex Shape
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We revisit the problem of building weak -nets for convex ranges over a point set in ^d. Unfortunately, the known constructions of weak -nets yields sets that are of size Ω(^-d e^c d^2 )., where c is some constant. We offer two alternative schemes that yield significantly smaller sets, and two related results, as follows: (A) Let be a sample of size (d^2 ^-1) points from the original point set (which is no longer needed), where hides polylogarithmic terms. Given a convex body , via a separation oracle, the algorithm performs a small sequence of (oracle) stabbing queries (computed from ) -- if none of the query points hits , then contains less than an -fraction of the input points. The number of stabbing queries performed is O( d^2 ^-1), and the time to compute them is (d^9 ^-1). To the best of our knowledge, this is the first weak -net related construction where all constants/bounds are polynomial in the dimension. (B) If one is allowed to expand the convex range before checking if it intersects the sample, then a sample of size .(^-(d+1)/2), from the original point set, is sufficient to form a net. (C) We show a construction of weak -nets which have the following additional property: For a heavy body, there is a net point that stabs the body, and it is also a good centerpoint for the points contained inside the body. (D) We present a variant of a known algorithm for approximating a centerpoint, improving the running time from (d^9) to (d^7). Our analysis of this algorithm is arguably cleaner than the previous version.
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