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.
KeywordsPlanar Graph Blue Point Colored Point Planar Drawing Convex Position
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