Odds-On Trees

TL;DR

Odds-on trees are a new data structure that efficiently filters queries based on a distribution, achieving near-optimal expected query times for various geometric problems.

Contribution

Introduction of odds-on trees, a compact data structure that improves distribution-sensitive query performance in geometric search problems.

Findings

Expected query time is O(H*+1) for queries drawn from distribution D.

Size of odds-on trees is O(n^ε), significantly smaller than the original data structure.

Applicable to a wide range of geometric searching problems in R^d.

Abstract

Let R^d -> A be a query problem over R^d for which there exists a data structure S that can compute P(q) in O(log n) time for any query point q in R^d. Let D be a probability measure over R^d representing a distribution of queries. We describe a data structure called the odds-on tree, of size O(n^\epsilon) that can be used as a filter that quickly computes P(q) for some query values q in R^d and relies on S for the remaining queries. With an odds-on tree, the expected query time for a point drawn according to D is O(H*+1), where H* is a lower-bound on the expected cost of any linear decision tree that solves P. Odds-on trees have a number of applications, including distribution-sensitive data structures for point location in 2-d, point-in-polytope testing in d dimensions, ray shooting in simple polygons, ray shooting in polytopes, nearest-neighbour queries in R^d, point-location in arrangements of hyperplanes in R^d, and many other geometric searching problems that can be solved in the linear-decision tree model. A standard lifting technique extends these results to algebraic decision trees of constant degree. A slightly different version of odds-on trees yields similar results for orthogonal searching problems that can be solved in the comparison tree model.