The complexity of the union of (α,β)-covered objects

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35 Scopus citations


An (α, β)-covered object is a simply connected planar region c with the property that for each point p ∈ ∂c there exists a triangle contained in c and having p as a vertex, such that all its angles are at least α > 0 and all its edges are at least β.diam(c)-long. This notion extends that of fat convex objects. We show that the combinatorial complexity of the union of n (α, β)-covered objects of "constant description complexity" is O(λ s+2(n) log 2 n log log n), where s is the maximum number of intersections between the boundaries of any pair of given objects, and λ s (n) denotes the maximum length of an (n, s)-Davenport-Schinzel sequence. Our result extends and improves previous results concerning convex α-fat objects.

Original languageEnglish (US)
Pages (from-to)775-787
Number of pages13
JournalSIAM Journal on Computing
Issue number4
StatePublished - 2005


  • Fat objects
  • Motion planning
  • Realistic input models

ASJC Scopus subject areas

  • Computer Science(all)
  • Mathematics(all)


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