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 boolean OR B and alternates colors. When these points are collinear, we describe a Theta(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. (C) 2016 Elsevier B.V. All rights reserved.