Epsilon-nets and simplex range queries

Abstract

We present a new technique for half-space and simplex range query using O(n) space and O(na) query time, where α 0. These bounds are better than those previously published for all d ≥ 2. The technique uses random sampling to build a partition-tree structure. We introduce the concept of an ϵ-net for an abstract set of ranges to describe the desired result of this random sampling and give necessary and sufficient conditions that a random sample is an ϵ-net with high probability. We illustrate the application of these ideas to other range query problems.