Odds-On Trees

Let P : Rd → A be a query problem over Rd for which there exists a data structure S that can compute P(q) in O(log n) time for any query point q ∈ Rd. Let D be a probability measure over Rd representing a distribution of queries. We describe a data structure T = TP,D, called the odds-on tree, of size O(n ) that can be used as a filter that quickly computes P(q) for some query values in Rd 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 Rd, pointlocation in arrangements of hyperplanes in Rd, 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.