Nearly Optimal Expected-Case Planar Point Location

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Nearly Optimal Expected-Case Planar Point Location (2000)

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by Sunil Arya Theocharis , Sunil Arya , Theocharis Malamatos , David M. Mount

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@MISC{Theocharis00nearlyoptimal,
    author = {Sunil Arya Theocharis and Sunil Arya and Theocharis Malamatos and David M. Mount},
    title = {Nearly Optimal Expected-Case Planar Point Location},
    year = {2000}
}

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Abstract

We consider the planar point location problem from the perspective of expected search time. We are given a planar polygonal subdivision S and for each polygon of the subdivision the probability that a query point lies within this polygon. The goal is to compute a search structure to determine which cell of the subdivision contains a given query point, so as to minimize the expected search time. This is a generalization of the classical problem of computing an optimal binary search tree for one-dimensional keys. In the one-dimensional case it has long been known that the entropy H of the distribution is the dominant term in the lower bound on the expected-case search time, and further there exist search trees achieving expected search times of at most H +2. Prior to this work, there has been no known structure for planar point location with an expected search time better than 2H, and this result required strong assumptions on the nature of the query point distribution. Here we present a data structure whose expected search time is nearly equal to the entropy lower bound, namely H + o(H). The result holds for any polygonal subdivision in which the number of sides of each of the polygonal cells is bounded, and there are no assumptions on the query distribution within each cell. We extend these results to subdivisions with convex cells, assuming a uniform query distribution within each cell.

Keyphrases

search time optimal expected-case planar point location query point search tree search structure one-dimensional case one-dimensional key known structure classical problem data structure strong assumption polygonal subdivision optimal binary search tree uniform query distribution query point distribution planar polygonal subdivision dominant term convex cell planar point location polygonal cell query distribution expected-case search time planar point location problem

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