k-Sets of Convex Inclusion Chains of Planar Point Sets

  • Wael El Oraiby
  • Dominique Schmitt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)


Given a set V of n points in the plane, we introduce a new number of k-sets that is an invariant of V: the number of k-sets of a convex inclusion chain of V. A convex inclusion chain of V is an ordering (v 1, v 2, ..., v n ) of the points of V such that no point of the ordering belongs to the convex hull of its predecessors. The k-sets of such a chain are then the distinct k-sets of all the subsets {v 1, ..., v i }, for all i in {k+1, ..., n}. We show that the number of these k-sets depends only on V and not on the chosen convex inclusion chain. Moreover, this number is surprisingly equal to the number of regions of the order-k Voronoi diagram of V. As an application, we give an efficient on-line algorithm to compute the k-sets of the vertices of a simple polygonal line, no vertex of which belonging to the convex hull of its predecessors on the line.


Convex Hull Polygonal Line Oriented Edge Planar Point Counter Clockwise Direction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Wael El Oraiby
    • 1
  • Dominique Schmitt
    • 1
  1. 1.Laboratoire MIAUniversité de Haute-AlsaceMulhouse CedexFrance

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