Two Heuristics for the Euclidean Steiner Tree Problem

The Euclidean Steiner tree problem is to find the tree with minimal Euclidean length spanning a set of fixed points in the plane, allowing the addition of auxiliary points to the set (Steiner points). The problem is NP-hard, so polynomial-time heuristics are desired. We present two such heuristics, both of which utilize an efficient method for computing a locally optimal tree with a given topology. The first systematically inserts Steiner points between edges of the minimal spanning tree meeting at angles less than 120 degrees, performing a local optimization at the end. The second begins by finding the Steiner tree for three of the fixed points. Then, at each iteration, it introduces a new fixed point to the tree, connecting it to each possible edge by inserting a Steiner point, and minimizes over all connections, performing a local optimization for each. We present a variety of test cases that demonstrate the strengths and weaknesses of both algorithms. 1 Steiner Trees Given a set of fixed (or terminal) points t1, . . . , tn in < , the Steiner tree problem is to find the shortest network (in the Euclidean sense) connecting School of Computer Science, Carnegie Mellon University, Pittsburgh, PA. E-mail: [email protected]. The work of this author was conducted at New York University with support from National Science Foundation grant CCR-9625955. Computer Science Department, Courant Institute of Mathematical Sciences, New York University, New York, NY. E-mail: [email protected]. The work of this author was supported in part by National Science Foundation grant CCR-9625955.