TY - JOUR
T1 - Approximate polytope membership queries
JF - Proceedings of 43rd Annual ACM Symposium on Theory of Computing
Y1 - 2011
A1 - Arya,S.
A1 - da Fonseca,G. D
A1 - Mount, Dave
AB - We consider an approximate version of a fundamental geo-metric search problem, polytope membership queries. Given a convex polytope P in Rd, presented as the intersection of halfspaces, the objective is to preprocess P so that, given a query point q, it is possible to determine efficiently whether q lies inside P subject to an error bound ε. Previous solutions to this problem were based on straight- forward applications of classic polytope approximation tech- niques by Dudley (1974) and Bentley et al. (1982). The former yields minimum storage, and the latter yields con- stant query time. A space-time tradeoff can be obtained by interpolating between the two. We present the first sig- nificant improvements to this tradeoff. For example, using the same storage as Dudley, we reduce the query time from O(1/ε (d−1)/2 ) to O(1/ε (d−1)/4 ). Our approach is based on a very simple algorithm. Both lower bounds and upper bounds on the performance of the algorithm are presented. To establish the relevance of our results, we introduce a reduction from approximate nearest neighbor searching to approximate polytope membership queries. We show that our tradeoff provides significant improvements to the best known space-time tradeoffs for approximate nearest neigh- bor searching. Furthermore, this is achieved with construc- tions that are much simpler than existing methods
ER -