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Combinatorial Geometry (1995)

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by P. K. Agarwal
Citations:183 - 23 self
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BibTeX

@MISC{Agarwal95combinatorialgeometry,
    author = {P. K. Agarwal},
    title = {Combinatorial Geometry},
    year = {1995}
}

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Abstract

Abstract. Let P be a set of n points in ~d (where d is a small fixed positive integer), and let F be a collection of subsets of ~d, each of which is defined by a constant number of bounded degree polynomial inequalities. We consider the following F-range searching problem: Given P, build a data structure for efficient answering of queries of the form, &quot;Given a 7 ~ F, count (or report) the points of P lying in 7.&quot; Generalizing the simplex range searching techniques, we give a solution with nearly linear space and preprocessing time and with O(n 1- x/b+~) query time, where d < b < 2d- 3 and ~> 0 is an arbitrarily small constant. The acutal value of b is related to the problem of partitioning arrangements of algebraic surfaces into cells with a constant description complexity. We present some of the applications of F-range searching problem, including improved ray shooting among triangles in ~3 1.

Keyphrases

combinatorial geometry    constant number    data structure    small fixed positive integer    constant description complexity    small constant    acutal value    bounded degree polynomial inequality    algebraic surface    linear space    ray shooting    query time    following f-range    simplex range    efficient answering    preprocessing time   

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