Three Evolutionary Codings of Rectilinear Steiner Arborescences

A rectilinear Steiner arborescence connects points in the Euclidean plane’s first quadrant and the origin with directed rectilinear edges from the origin up and to the right. The search for arborescences of minimum total length is NP-hard and finds applications in VLSI design. A greedy heuristic for this problem often returns near-optimum arborescences. Three genetic algorithms encode candidate arborescences as permutations of the points, as perturbations of the points’ locations, and as perturbations of the points’ rectilinear distances from the origin. In comparisons on twenty collections of 50 to 250 points in the first quadrant, the permutation-coded GA returns arborescences that are longer than those of the greedy heuristic. The two perturbation-coded GAs return nearly identical results, their arborescences are consistently shorter than those of the heuristic, and they preserve their advantage as the number of points grows. These results support the usefulness of perturbation codings in evolutionary algorithms for geometric problems like the search for short rectilinear Steiner arborescences.