Lower bounds for expected-case planar point location

13 years 5 months ago

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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 pz 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. Further it is known that it is possible to construct a data structure that answers point-location queries in H + 2 2H + 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 there exists a query distibution Q over S such that even when we are given complete information on Q, the optimal expected query time must be at least H + ( H), which differs just by a constant factor in the second order term from the best known upper bound.

Theocharis Malamatos

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