Odds-On Trees

Abstract

Let P : 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 ∈ R d . Let D be a probability measure over R d representing a distribution of queries. We describe a data structure T = T P,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 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 , pointlocation 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.

Adaptive Planar Point Location

“…Fins and their triangulations were first described by Hershberger and Suri [13] in the context of solving the ray-shooting problem in a simple polygon. 5 They give a tighter but more involved analysis that yields an O(log |α|) bound on the triangles intersected by a segment. For our purposes, a polylogarithmic bound suffices.…”

“…Each decision tree node contains a linear function which exactly models a point-line comparison. Given a connected subdivision S with n vertices and the query distribution, Collete et al [10] designed a structure that uses O(n) space and answers a query in O(H * ) expected time, where H * is the minimum expected time needed by any point location linear decision tree for S. When the query distribution is given, Afshani, Barbay, and Chan [1] and Bose et al [5] solved several geometric query problems, including planar point location in general subdivisions, with optimal expected performance with respect to linear decision trees.…”

A static optimality transformation with applications to planar point location

“…Recently, several data structures have been developed for optimal point location where the distribution is known in advance for convex connected [17], connected [18], and arbitrary polygonal [14] subdivisions of the plane, as well as the more general odds-on trees [13]. Unfortunately, these structures are not biased according to our definition, since entropy-based lower bounds are not meaningful for them: a convex k-gon splits the plane into two regions, so the entropy of the query outcomes is constant.…”

Dynamic Distribution-Sensitive Point Location

“…Later, Collete et al [15] obtained the same result for connected subdivisions. So did Afshani et al [2] and Bose et al [7] for general subdivisions.…”