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Approximate polytope membership queries. (English) Zbl 1381.52019

Summary: 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 R. M. Dudley [J. Approx. Theory 10, 227–236 (1974; Zbl 0275.41011)] and J. L. Bentley et al. [Commun. ACM 25, 64–68 (1982; Zbl 0466.68059)]. 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 \(\alpha\geq 4\), 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.

MSC:

52B11 \(n\)-dimensional polytopes
68W25 Approximation algorithms
68P05 Data structures
52A27 Approximation by convex sets
41A45 Approximation by arbitrary linear expressions
41A63 Multidimensional problems
26A51 Convexity of real functions in one variable, generalizations
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