On orientation metric and Euclidean Steiner tree constructions

Abstract

We consider Steiner minimal trees (SMT) in the plane, where only orientations with angle i/spl pi///spl sigma/, 0/spl les/i/spl les//spl sigma/-1 and a an integer, are allowed. The orientations define a metric, called the orientation metric, /spl lambda//sub /spl sigma//, in a natural way. In particular, /spl lambda//sub 2/ metric is the rectilinear metric and the Euclidean metric can be regarded as /spl lambda//sub /spl infin// metric. In this paper, we provide a method to find an optimal /spl lambda//sub /spl sigma// SMT for 3 or 4 points by analyzing the topology of /spl lambda//sub /spl sigma// SMT's in great detail. Utilizing these results and based on the idea of loop detection, we further develop an O(n/sup 2/) time heuristic for the general /spl lambda//sub /spl sigma// SMT problem, including the Euclidean metric. Experiments performed on publicly available benchmark data for 12 different metrics, plus the Euclidean metric, demonstrate the efficiency of our algorithms and the quality of our results.