Communication Dans Un Congrès Année : 2018

Approximate Convex Intersection Detection with Applications to Width and Minkowski Sums

Résumé

Approximation problems involving a single convex body in d-dimensional space have received a great deal of attention in the computational geometry community. In contrast, works involving multiple convex bodies are generally limited to dimensions d3 and/or do not consider approximation. In this paper, we consider approximations to two natural problems involving multiple convex bodies: detecting whether two polytopes intersect and computing their Minkowski sum. Given an approximation parameter ε>0, we show how to independently preprocess two polytopes A,B into data structures of size O(1/ε(d1)/2) such that we can answer in polylogarithmic time whether A and B intersect approximately. More generally, we can answer this for the images of A and B under affine transformations. Next, we show how to ε-approximate the Minkowski sum of two given polytopes defined as the intersection of n halfspaces in O(nlog(1/ε)+1/ε(d1)/2+α) time, for any constant α>0. Finally, we present a surprising impact of these results to a well studied problem that considers a single convex body. We show how to ε-approximate the width of a set of n points in O(nlog(1/ε)+1/ε(d1)/2+α) time, for any constant α>0, a major improvement over the previous bound of roughly O(n+1/εd1) time.
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Dates et versions

hal-01890039 , version 1 (10-10-2018)

Identifiants

Citer

Sunil Arya, Guilherme D. da Fonseca, David M. Mount. Approximate Convex Intersection Detection with Applications to Width and Minkowski Sums. ESA 2018 - European Symposium on Algorithms, Aug 2018, Helsinki, Finland. ⟨10.4230/LIPIcs.ESA.2018.3⟩. ⟨hal-01890039⟩
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