A simple entropy-based algorithm for planar point location
ACM Transactions on Algorithms (TALG)
Lower bounds for expected-case planar point location
Computational Geometry: Theory and Applications
Distribution-sensitive point location in convex subdivisions
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Self-improving algorithms for delaunay triangulations
Proceedings of the twenty-fourth annual symposium on Computational geometry
Succinct geometric indexes supporting point location queries
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
A static optimality transformation with applications to planar point location
Proceedings of the twenty-seventh annual symposium on Computational geometry
SIAM Journal on Computing
Succinct geometric indexes supporting point location queries
ACM Transactions on Algorithms (TALG)
Entropy, triangulation, and point location in planar subdivisions
ACM Transactions on Algorithms (TALG)
A self-adjusting data structure for multidimensional point sets
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Distance-Sensitive planar point location
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
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Point location is the problem of preprocessing a planar polygonal subdivision $S$ of size $n$ into a data structure in order to determine efficiently the cell of the subdivision that contains a given query point. We consider this problem from the perspective of expected query time. We are given the probabilities $p_z$ that the query point lies within each cell $z \in S$. The entropy $H$ of the resulting discrete probability distribution is the dominant term in the lower bound on the expected-case query time. We show that it is possible to achieve query time $H + O(\sqrt{H}+1)$ with space $O(n)$, which is optimal up to lower order terms in the query time. We extend this result to subdivisions with convex cells, assuming a uniform query distribution within each cell. In order to achieve space efficiency, we introduce the concept of entropy-preserving cuttings.