Planar point location using persistent search trees
Communications of the ACM
Optimal point location in a monotone subdivision
SIAM Journal on Computing
An O(n log n) algorithm for the all-nearest-neighbors problem
Discrete & Computational Geometry
Simplified linear-time Jordan sorting and polygon clipping
Information Processing Letters
A fast planar partition algorithm, I
Journal of Symbolic Computation
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Proceedings of the twelfth annual symposium on 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
Expected-case complexity of approximate nearest neighbor searching
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
A new point-location algorithm and its practical efficiency: comparison with existing algorithms
ACM Transactions on Graphics (TOG)
Parallel Construction of Quadtrees and Quality Triangulations
WADS '93 Proceedings of the Third Workshop on Algorithms and Data Structures
A static optimality transformation with applications to planar point location
Proceedings of the twenty-seventh annual symposium on Computational geometry
Entropy, triangulation, and point location in planar subdivisions
ACM Transactions on Algorithms (TALG)
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Planar point location is among the most fundamental search problems in computational geometry. Although this problem has been heavily studied from the perspective of worst-case query time, there has been surprisingly little theoretical work on expected-case query time. We are given an n-vertex planar polygonal subdivision S satisfying some weak assumptions (satisfied, for example, by all convex subdivisions). We are to preprocess this into a data structure so that queries can be answered efficiently. We assume that the two coordinates of each query point are generated independently by a probability distribution also satisfying some weak assumptions (satisfied, for example, by the uniform distribution). In the decision tree model of computation, it is well-known from information theory that a lower bound on the expected number of comparisons is entropy(S). We provide two data structures, one of size O(n2) that can answer queries in 2 entropy(S) + O(1) expected number of comparisons, and another of size O(n) that can answer queries in (4 + O(1/√log n)) entropy(S)+O(1) expected number of comparisons. These structures can be built in O(n2) and O(n log n) time respectively. Our results are based on a recent result due to Arya and Fu, which bounds the entropy of overlaid subdivisions.