Biased Range Trees

TL;DR

This paper introduces biased range trees, a data structure that efficiently preprocesses point sets for range counting queries with expected query times close to optimal, using O(n log n) space and preprocessing.

Contribution

It presents biased range trees that adapt to query distributions, achieving near-optimal expected query times for 2D orthogonal range counting.

Findings

Expected query time matches optimal decision tree within a constant factor.

Memory and preprocessing requirements are O(n log n).

The data structure effectively adapts to query distributions.

Abstract

A data structure, called a biased range tree, is presented that preprocesses a set S of n points in R^2 and a query distribution D for 2-sided orthogonal range counting queries. The expected query time for this data structure, when queries are drawn according to D, matches, to within a constant factor, that of the optimal decision tree for S and D. The memory and preprocessing requirements of the data structure are O(n log n).