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Alternating Paths and Cycles of Minimum Length

  • William S. Evans
  • Giuseppe Liotta
  • Henk Meijer
  • Stephen WismathEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)

Abstract

Let R be a set of n red points and B be a set of n blue points in the Euclidean plane. We study the problem of computing a planar drawing of a cycle of minimum length that contains vertices at points \(R \cup B\) and alternates colors. When these points are collinear, we describe a \(\varTheta (n \log n)\)-time algorithm to find such a shortest alternating cycle where every edge has at most two bends. We extend our approach to compute shortest alternating paths in \(O(n^2)\) time with two bends per edge and to compute shortest alternating cycles on 3-colored point-sets in \(O(n^2)\) time with O(n) bends per edge. We also prove that for arbitrary k-colored point-sets, the problem of computing an alternating shortest cycle is NP-hard, where k is any positive integer constant.

Keywords

Planar Graph Blue Point Colored Point Planar Drawing Convex Position 
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 2015

Authors and Affiliations

  • William S. Evans
    • 1
  • Giuseppe Liotta
    • 2
  • Henk Meijer
    • 3
  • Stephen Wismath
    • 4
    Email author
  1. 1.University of British ColumbiaVancouverCanada
  2. 2.Universitá degli Studi di PerugiaPerugiaItaly
  3. 3.U. C. RooseveltMiddelburgThe Netherlands
  4. 4.University of LethbridgeLethbridgeCanada

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