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)


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.


Planar Graph Blue Point Colored Point Planar Drawing Convex Position 
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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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