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

Pankaj K. Agarwal, Alon Efrat, Micha Sharir

Research output: Contribution to journalArticle

71 Citations (Scopus)

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+ε ψ(n/k)) for any ε > 0, where ψ(r) is the maximum complexity of the lower envelope of a subset of at most r functions of F. This bound is nearly optimal in the worst case and implies the existence of shallow cuttings, in the sense of [J. Matousek, Comput. Geom., 2 (1992), pp. 169-186], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including (i) data structures for several generalized 3-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)
Pages (from-to)912-953
Number of pages42
JournalSIAM Journal on Computing
Volume29
Issue number3
StatePublished - Dec 1999
Externally publishedYes

Fingerprint

Algebraic function
Arrangement
Vertical
Lower Envelopes
Range Searching
Geometric Optimization
Dynamic Data Structures
Decomposition
Combinatorial Complexity
Decompose
Distance Function
Maximum Degree
Data structures
Euclidean
Data Structures
Efficient Algorithms
Optimization Problem
Imply
Subset
Set theory

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Applied Mathematics
  • Theoretical Computer Science

Cite this

Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. / Agarwal, Pankaj K.; Efrat, Alon; Sharir, Micha.

In: SIAM Journal on Computing, Vol. 29, No. 3, 12.1999, p. 912-953.

Research output: Contribution to journalArticle

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