Elements of information theory
Elements of information theory
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
The art of computer programming, volume 3: (2nd ed.) sorting and searching
The art of computer programming, volume 3: (2nd ed.) sorting and searching
Handbook of discrete and computational geometry
Methods for achieving fast query times in point location data structures
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
On the exact worst case query complexity of planar point location
Journal of Algorithms
Entropy-preserving cuttings and space-efficient planar point location
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Nearly optimal expected-case planar point location
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Expected asymptotically optimal planar point location
Computational Geometry: Theory and Applications - Special issue on the 10th fall workshop on computational geometry
A simple entropy-based algorithm for planar point location
ACM Transactions on Algorithms (TALG)
Optimal Expected-Case Planar Point Location
SIAM Journal on Computing
Practical distribution-sensitive point location in triangulations
Computer Aided Geometric Design
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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 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.