On range searching with semialgebraic sets

Journal Article

Let P be a set of n points in ℝ d (where d is a small fixed positive integer), and let Γ be a collection of subsets of ℝ d, each of which is defined by a constant number of bounded degree polynomial inequalities. We consider the following Γ-range searching problem: Given P, build a data structure for efficient answering of queries of the form, "Given a γ∈Γ, count (or report) the points of P lying in γ." Generalizing the simplex range searching techniques, we give a solution with nearly linear space and preprocessing time and with O(n 1-1/b+δ ) query time, where d≤b≤2d-3 and δ>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 Γ-range searching problem, including improved ray shooting among triangles in ℝ3. © 1994 Springer-Verlag New York Inc.

Full Text

Duke Authors

Cited Authors

  • Agarwal, PK; Matousek, J

Published Date

  • December 1, 1994

Published In

Volume / Issue

  • 11 / 1

Start / End Page

  • 393 - 418

Electronic International Standard Serial Number (EISSN)

  • 1432-0444

International Standard Serial Number (ISSN)

  • 0179-5376

Digital Object Identifier (DOI)

  • 10.1007/BF02574015

Citation Source

  • Scopus