Alternating Paths and Cycles of Minimum Length

Evans, William S. and Liotta, Giuseppe and Meijer, Henk and Wismath, Stephen (2015) Alternating Paths and Cycles of Minimum Length. In: Graph Drawing and Network Visualization: 23rd International Symposium, GD 2015, September 24-26, 2015, Los Angeles, CA, USA , pp. 383-394 (Official URL: http://dx.doi.org/10.1007/978-3-319-27261-0_32).

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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∪B and alternates colors. When these points are collinear, we describe a Θ(nlogn)-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.

Item Type:Conference Paper
Classifications:G Algorithms and Complexity > G.210 Bends
G Algorithms and Complexity > G.490 Embeddings
G Algorithms and Complexity > G.560 Geometry
P Styles > P.540 Planar
Z Theory > Z.250 Geometry
ID Code:1504

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