NDLI: Optimal approximate polytope membership

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Universal shape replicators via self-assembly with attractive and repulsive forces

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Optimal approximate polytope membership

Content Provider	ACM Digital Library
Author	Mount, David M. Arya, Sunil da Fonseca, Guilherme D.
Abstract	In the polytope membership problem, a convex polytope K in $ℝ^{d}$ is given, and the objective is to preprocess K into a data structure so that, given a query point q ∈ $ℝ^{d},$ it is possible to determine efficiently whether q ∈ K. We consider this problem in an approximate setting and assume that d is a constant. Given an approximation parameter ε > 0, the query can be answered either way if the distance from q to K's boundary is at most ε times K's diameter. Previous solutions to the problem were on the form of a space-time trade-off, where logarithmic query time demands $O(1/ε^{d™1})$ storage, whereas storage $O(1/ε^{(d™1)/2})$ admits roughly $O(1/ε^{(d™1)/8})$ query time. In this paper, we present a data structure that achieves logarithmic query time with storage of only $O(1/ε^{(d™1)/2}),$ which matches the worst-case lower bound on the complexity of any ε-approximating polytope. Our data structure is based on a new technique, a hierarchy of ellipsoids defined as approximations to Macbeath regions. As an application, we obtain major improvements to approximate Euclidean nearest neighbor searching. Notably, the storage needed to answer ε-approximate nearest neighbor queries for a set of n points in O(log n/ε) time is reduced to $O(n/ε^{d/2}).$ This halves the exponent in the ε-dependency of the existing space bound of roughly $O(n/ε^{d}),$ which has stood for 15 years (Har-Peled, 2001).
Starting Page	270
Ending Page	288
Page Count	19
File Format	PDF
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2017-01-16
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article

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