Worst-case minimum rectilinear steiner trees in all dimensions

Abstract

It is proved that the length of the longest possible minimum rectilinear Steiner tree of n points in the unit d-cube is asymptotic to βdn(d-1)/d, where βd is a constant that depends on the dimension d≥2. A method of Chung and Graham (1981) is generalized to dimension d to show that 1≤βd≤d4(1-d)/d. In addition to replicating Chung and Graham's exact determination of β2=1, this generalization yields new bounds such as 1≤β3<1.191 and {Mathematical expression}. © 1992 Springer-Verlag New York Inc.