Alternating Paths and Cycles of Minimum Length
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.
- 4.Bastert, O., Fekete, S.P.: Geometrische Verdrahtungsprobleme. Technical Report 96–247, Universität zu Köln (1996)Google Scholar
- 12.Kaneko, A., Kano, M., Suzuki, K.: Path coverings of two sets of points in the plane. In: Pach, J. (ed.) Towards a Theory of Geometric Graph, vol. 342. American Mathematical Society, Providence (2004)Google Scholar
- 13.Kaneko, A., Kano, M., Tokunaga, S.: Straight-line embeddings of three rooted trees in the plane. In: Canadian Conference on Computational Geometry, CCCG 1998 (1998)Google Scholar