Approximate polytope membership queries

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 any query point q ∈ R d , it is possible to determine efficiently whether q ∈ K. We consider this problem in an approximate setting. Given an approximation parameter ε, the query can be answered either way if the distance from q to K's boundary is at most ε 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 (1974) and Bentley et al. (1982). 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 α ≥ 4, this data structure achieves query time roughly O 1/ε (d−1)/α and space roughly O 1/ε (d−1)(1−Ω(log α)/α) . 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 Ω 1/εOur 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 R d . For example, we show that it is possible to achieve a query time of roughly O(log n + 1/ε d/4 ) with space roughly O(n/ε d/4 ), thus reducing by half the exponent in the space bound.

Approximate Nearest Line Search in High Dimensions

“…We also note that in low dimensions several approximate algorithms for general polytope membership are known [5]. However, those algorithms have query time and space exponential in the dimension and therefore, our results are specially interesting in high dimensions.…”

Products of Euclidean Metrics, Applied to Proximity Problems among Curves

“…Efficient deterministic solutions exist when the dimension is bounded, e.g. [Arya et al 2011[Arya et al , 2009Har-Peled et al 2012] and are based on a notion of Approximate Voronoi Diagrams (AVDs) or tree-based data structures, while for high-dimensional data the state-ofthe-art solutions are mainly based either on the concept of Locality Sensitive Hashing (LSH), e.g. [Andoni et al 2017;Har-Peled et al 2012], or on random projections, e.g.…”

Proximity problems for high-dimensional data

“…They are implemented on a hierarchical quadtree-based subdivision of space into cells, each storing a number of representative points, such that for any query point lying in the cell, at least one of the representatives is an approximate nearest neighbor. Further improvements to the spacetime trade offs for ANN are obtained in [14].…”