Approximate Euclidean Steiner trees

An approximate Steiner tree is a Steiner tree on a given set of terminals in Euclidean space such that the angles at the Steiner points are within a specified error e from 120 degrees. This notion arises in numerical approximations of minimum Steiner trees (W. D. Smith, Algorithmica, 7 (1992), 137–177). We investigate the worst-case relative error of the length of an approximate Steiner tree compared to the shortest tree with the same topology. Rubinstein, Weng and Wormald (J. Global Optim. 35 (2006), 573–592) conjectured that this relative error is at most linear in e, independent of the number of terminals. We verify their conjecture for the two-dimensional case as long as the error e is sufficiently small in terms of the number of terminals. We derive a lower bound linear in e for the relative error in the two-dimensional case when e is sufficiently small in terms of the number of terminals. We find improved estimates of the relative error for larger values of e, and calculate exact values in the plane for three and four terminals.