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Efficient Expected-Case Algorithms for Planar Point Location

Arya, Sunil and Cheng, Siu-Wing and Mount, David M and Ramesh, H (2000) Efficient Expected-Case Algorithms for Planar Point Location. In: 7th Scandinavian Workshop on Algorithm Theory- SWAT 2000, July 2000, Bergen, Norway, pp. 353-366.

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Abstract

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(n(2)) 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/root log n)) entropy(S) + O(1) expected number of comparisons. These structures can be built in O(n(2)) 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.

Item Type: Conference Paper
Additional Information: Copyright for this article belongs to Springer Verlag.
Department/Centre: Division of Electrical Sciences > Computer Science & Automation (Formerly, School of Automation)
Date Deposited: 27 Sep 2004
Last Modified: 11 Jan 2012 08:37
URI: http://eprints.iisc.ernet.in/id/eprint/1672

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