On levels in arrangements of curves

Research output: Contribution to journalArticlepeer-review

Abstract

Analyzing the worst-case complexity of the k-level in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O(nk1-2/3(s))) for curves that are graphs of polynomial functions of an arbitrary fixed degree s. Previously, nontrivial results were known only for the case s = 1 and s = 2. We also improve the earlier bound for pseudo-parabolas (curves that pairwise intersect at most twice) to O(nk7/9 log2/3 k). The proofs are simple and rely on a theorem of Tamaki and Tokuyama on cutting pseudo-parabolas into pseudo-segments, as well as a new observation for cutting pseudo-segments into pieces that can be extended to pseudo-lines. We mention applications to parametric and kinetic minimum spanning trees.

Original languageEnglish (US)
Pages (from-to)219-227
Number of pages9
JournalAnnual Symposium on Foundations of Computer Science - Proceedings
StatePublished - 2000
Externally publishedYes

ASJC Scopus subject areas

  • Hardware and Architecture

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