A scalable genetic algorithm for the rectilinear Steiner problem

Abstract

The rectilinear Steiner problem seeks the shortest tree made up of horizontal and vertical line segments that connects a set of points in the plane. The extra points where the segments meet are called Steiner points. Evolutionary algorithms for this problem have encoded rectilinear Steiner trees by extending codings of spanning trees to specify Steiner point choices for the spanning tree edges. These algorithms have been slow and have performed poorly on larger problem instances. The genetic algorithm presented here searches only the space of Steiner point assignments to the edges of a minimum rectilinear spanning tree. In tests on 45 instances of the rectilinear Steiner problem, it returns good, though never optimal, trees. The algorithm scales well; it evaluates chromosomes in time that is linear in the number of points, and its performance does not deteriorate as that number increases.