Algorithms for Polytope Covering and Approximation, and for Approximate Closest-point Queries

This paper gives an algorithm for polytope covering : let L and U be sets of points in R, comprising n points altogether. A cover for L from U is a set C ⊂ U with L a subset of the convex hull of C. Suppose c is the size of a smallest such cover, if it exists. The randomized algorithm given here finds a cover of size no more than c(5d ln c), for c large enough. The algorithm requires O(cn) expected time. More exactly, the time bound is O(cn + c(nc)), where γ ≡ 1/bd/2c. The previous best bounds were cO(logn) cover size in O(n) time.[MS92b] A variant algorithm is applied to the problem of approximating the boundary of a polytope with the boundary of a simpler polytope. For an appropriate measure, an approximation with error requires c = O(1/ )(d−1)/2 vertices, and the algorithm gives an approximation with c(5d ln(1/ )) vertices. The algorithms apply ideas previously used for small-dimensional linear programming. The final result here applies polytope approximation to the the post office problem: given n points (called sites) in d dimensions, build a data structure so that given a query point q, the closest site to q can be found quickly. The algorithm given here is given also a relative error bound , and depends on a ratio ρ, which is no more than the ratio of the distance between the farthest pair of sites to the distance between the closest pair of sites. The algorithm builds a data structure of size O(n(log ρ)/ d/2 in time O(n(log ρ))/ . With this data structure, closest-point queries can be answered in O(logn)/ d/2 time. ∗This manuscript merges [Cla93] and [Cla94] 1In this paper, δ will denote any fixed value greater than zero.