NDLI: Approximate Polytope Membership Queries

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Approximate Polytope Membership Queries

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Mount, David M. Arya, Sunil da Fonseca, Guilherme D.
Copyright Year	2018
Abstract	In the polytope membership problem, a convex polytope $K$ in $\mathbb{R}^d$ is given, and the objective is to preprocess $K$ into a data structure so that, given any query point $q \in \mathbb{R}^d$, it is possible to determine efficiently whether $q \in K$. We consider this problem in an approximate setting. Given an approximation parameter $\varepsilon$, the query can be answered either way if the distance from $q$ to $K$'s boundary is at most $\varepsilon$ times $K$'s diameter. We assume that the dimension $d$ is fixed, and $K$ is presented as the intersection of $n$ halfspaces. Previous solutions to approximate polytope membership were based on straightforward applications of classic polytope approximation techniques by Dudley [Approx. Theory, 10 (1974), pp. 227--236] and Bentley, Faust, and Preparata [Commun. ACM, 25 (1982), pp. 64--68]. The former is optimal in the worst case with respect to space, and the latter is optimal with respect to query time. We present four main results. First, we show how to combine the two above techniques to obtain a simple space-time trade-off. Second, we present an algorithm that dramatically improves this trade-off. In particular, for any constant $\alpha \ge 4$, this data structure achieves query time roughly $O(1/\varepsilon^{(d-1)/\alpha})$ and space roughly $O(1/\varepsilon^{(d-1)(1 - \Omega(\log \alpha)/\alpha)})$. We do not know whether this space bound is tight, but our third result shows that there is a convex body such that our algorithm achieves a space of at least $\Omega( 1/\varepsilon^{(d-1)(1-O(\sqrt{\alpha})/\alpha} )$. Our fourth result shows that it is possible to reduce approximate Euclidean nearest neighbor searching to approximate polytope membership queries. Combined with the above results, this provides significant improvements to the best known space-time trade-offs for approximate nearest neighbor searching in $\mathbb{R}^d$. For example, we show that it is possible to achieve a query time of roughly $O(\log n + 1/\varepsilon^{d/4})$ with space roughly $O(n/\varepsilon^{d/4})$, thus reducing by half the exponent in the space bound.
Sponsorship	Conselho Nacional de Desenvolvimento Científico e Tecnológico. Research Grants Council, University Grants Committee. Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro. National Science Foundation
Starting Page	1
Ending Page	51
Page Count	51
File Format	PDF
ISSN	00975397
DOI	10.1137/16M1061096
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	1
Volume Number	47
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2018-01-02
Access Restriction	Subscribed
Subject Keyword	polytope membership space-time trade-offs Mahler volume geometric retrieval convex approximation Data structures Approximation algorithms approximation algorithms nearest neighbor searching
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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