The k-Steiner Ratio in the Rectilinear Plane

Ž . A Steiner minimum tree SMT in the rectilinear plane is the shortest length tree interconnecting a set of points, called the regular points, possibly using Ž . additional vertices. A k-size Steiner minimum tree kSMT is one that can be split into components where all regular points are leaves and all components have at most k leaves. The k-Steiner ratio in the rectilinear plane, r , is the infimum of k the ratios SMTrkSMT over all finite sets of regular points. The k-Steiner ratio is used to determine the performance ratio of several recent polynomial-time approximations for Steiner minimum trees. Previously it was known that in the rectilinear Ž . Ž . Ž . Ž . Ž . plane, r s 2r3, r s 4r5, and 2k y 2 r 2k y 1 F r L F 2k y 1 r 2k for 2 3 k 1 Ž k G 4. In 1991, P. Berman and V. Ramaiyer conjectured that in fact r s 2k y k . Ž . 1 r 2k for k G 4. In this paper we prove their conjecture. Q 1998 Academic Press