Approximate halfspace range counting

We present a simple scheme extending the shallow partitioning data structures of Matoušek, which supports efficient approximate halfspace range-counting queries in ℝ^d with relative error ε. Specifically, the problem is, given a set P of n points in ℝ^d, to preprocess them into a data structure that returns, for a query halfspace h, a number t so that (1 - ε)|h ∩ P| ≤ t ≤ (1 + ε)| h ∩P|. One of our data structures requires linear storage and O(n^1+δ) preprocessing time, for any δ > 0, and answers a query in time O(ε^-γ n^{1-1/⌊d/2⌋}2^b log^* n) for any γ > 2/⌊d/2⌋; the choice of γ and δ affects b and the implied constants. Several variants and extensions are also discussed. As presented, the construction of the structure is mostly deterministic, except for one critical randomized step, and so are the query, storage, and preprocessing costs. The quality of approximation, for every query, is guaranteed with high probability. The construction can also be fully derandomized, at the expense of increasing preprocessing time.