Epsilon-nets and simplex range queries

We present a new technique for half-space and simplex range query using <italic>&Ogr;</italic>(<italic>n</italic>) space and <italic>&Ogr;</italic>(<italic>n</italic><supscrpt><italic>a</italic></supscrpt>) query time, where <italic>a</italic> < <italic>d</italic>(d-1)/<italic>d</italic>(<italic>d</italic>-1) + 1 + γ for all dimensions <italic>d</italic> ≥ 2 and <italic>γ</italic> > 0. These bounds are better than those previously published for all <italic>d</italic> ≥ 2. The technique uses random sampling to build a partition-tree structure. We introduce the concept of an <italic>ε</italic>-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 <italic>ε</italic>-net with high probability. We illustrate the application of these ideas to other range query problems.