Lower bounds for expected-case planar point location

Abstract

Given a planar polygonal subdivision S, the point location problem is to preprocess S into a data structure so that the cell of the subdivision that contains a given query point can be reported efficiently. Suppose that we are given for each cell z ∈ S the probability p z that a query point lies in z. The entropy H of the resulting discrete probability distribution is a lower bound on the expected-case query time. In addition it is known that it is possible to construct a data structure that answers point-location queries in H + 2 2 H + o ( H ) expected number of comparisons. A fundamental question is how close to the entropy lower bound H the exact optimal expected query time can reach. In this paper we show that if only the probabilities p z are given and no information is available for the probability distribution within each cell, then the optimal expected query time must be at least H + H − O ( 1 ) . Further we show that there exists a query distribution Q over S such that even when we are given complete information on Q, the optimal expected query time must be at least H + 1 64 H − O ( 1 ) . Both these lower bounds differ just by a constant factor in the second order term from the best known upper bound.