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Reconstructing Point Set Order Typesfrom Radial Orderings

  • Oswin Aichholzer
  • Jean Cardinal
  • Vincent KustersEmail author
  • Stefan Langerman
  • Pavel Valtr
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8889)

Abstract

We consider the problem of reconstructing the combinatorial structure of a set of \(n\) points in the plane given partial information on the relative position of the points. This partial information consists of the radial ordering, for each of the \(n\) points, of the \(n-1\) other points around it. We show that this information is sufficient to reconstruct the chirotope, or labeled order type, of the point set, provided its convex hull has size at least four. Otherwise, we show that there can be as many as \(n-1\) distinct chirotopes that are compatible with the partial information, and this bound is tight. Our proofs yield polynomial-time reconstruction algorithms. These results provide additional theoretical insights on previously studied problems related to robot navigation and visibility-based reconstruction.

Keywords

Polynomial Time Convex Hull Radial System Local Sequence Order Type 
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 International Publishing Switzerland 2014

Authors and Affiliations

  • Oswin Aichholzer
    • 1
  • Jean Cardinal
    • 2
  • Vincent Kusters
    • 3
    Email author
  • Stefan Langerman
    • 2
  • Pavel Valtr
    • 4
  1. 1.Institute for Software TechnologyGraz University of TechnologyGrazAustria
  2. 2.Computer Science DepartmentUniversité libre de Bruxelles (ULB)BrusselsBelgium
  3. 3.Department of Computer ScienceETH ZürichZurichSwitzerland
  4. 4.Department of Applied MathematicsCharles UniversityPragueCzech Republic

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