Lower bounds for expected-case planar point location

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+22H+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+164H-O(1). Both these lower bounds differ just by a constant factor in the second order term from the best known upper bound.