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 ź B and alternates colors. When these points are collinear, we describe a ź ( n log ź 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.