Full Minimal Steiner Trees on Lattice Sets

Abstract

Given a finite set of points P in the Euclidean plane, the Steiner problem asks us to constuct a shortest possible network interconnecting P. Such a network is known as a minimal Steiner tree. The Steiner problem is an intrinsically difficult one, having been shown to be NP-hard [7]; however, it often proves far more tractable if we restrict our attention to points in special geometric configurations. One such restriction which has generated considerable interest is that of finding minimal Steiner trees for nice sets of integer lattice points. The first significant result in this direction was that of Chung and Graham [4], which, in effect, precisely characterized the minimal Steiner trees for any horizontal 2_n array of integer lattice points. In 1989, Chung et al. [3] examined a related problem, which they described as the Checkerboard Problem. They asked how to find a minimal Steiner tree for an n_n square lattice, that is, a collection of n_n points arranged in a regular lattice of unit squares like the corners of the cells of article no. TA962752