A new bound for the Steiner ratio

Abstract

Let V V denote a given set of n n points in the euclidean plane. A Steiner minimal tree for V V is the shortest network (clearly, it has to be a tree) interconnecting V V . Junctions of the network which are not in V V are called Steiner points (those in V V will be called regular points). A shortest tree interconnecting V V without using any Steiner points is called a minimal tree. Let σ ( V ) \sigma (V) and μ ( V ) \mu (V) denote the lengths of a Steiner minimal tree and a minimal tree, respectively. Define ρ \rho to be the greatest lower bound for the ratio σ ( V ) / μ ( V ) \sigma (V)/\mu (V) over all V V . We prove ρ > .8 \rho > .8 .