Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications

Pankaj K. Agarwal, Alon Efrat, Micha Sharir

Research output: Chapter in Book/Report/Conference proceedingConference contribution

22 Scopus citations

Abstract

Let F be a collection of n bivariate algebraic functions of constant maximum degree. We show that the combinatorial complexity of the vertical decomposition of the ≤k-level of the arrangement A(F) is O(k3+∈ψ(n/k)), for any ∈ > 0, where ψ(r) is the maximum complexity of the lower envelope of a subset of at most r functions of F. This result implies the existence of shallow cuttings, in the sense of [3, 31], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including: (i) data structures for several generalized three-dimensional range searching problems; (ii) dynamic data structures for planar nearest and farthest neighbor searching under various fairly general distance functions; (iii) an improved (near-quadratic) algorithm for minimum-weight bipartite Euclidean matching in the plane; and (iv) efficient algorithms for certain geometric optimization problems in static and dynamic settings.

Original languageEnglish (US)
Title of host publicationProceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995
PublisherAssociation for Computing Machinery
Pages39-50
Number of pages12
VolumePart F129372
ISBN (Electronic)0897917243
DOIs
StatePublished - Sep 1 1995
Externally publishedYes
Event11th Annual Symposium on Computational Geometry, SCG 1995 - Vancouver, Canada
Duration: Jun 5 1995Jun 7 1995

Other

Other11th Annual Symposium on Computational Geometry, SCG 1995
CountryCanada
CityVancouver
Period6/5/956/7/95

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Computational Mathematics

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    Agarwal, P. K., Efrat, A., & Sharir, M. (1995). Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. In Proceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995 (Vol. Part F129372, pp. 39-50). Association for Computing Machinery. https://doi.org/10.1145/220279.220284