Approximate Euclidean Steiner Trees

TL;DR

This paper studies the worst-case relative error in approximate Euclidean Steiner trees, confirming a conjecture for small errors in 2D and providing bounds and exact values for small terminal sets.

Contribution

It verifies a conjecture on the linear bound of relative error for small errors in 2D and derives new bounds and exact values for small terminal sets.

Findings

Conjecture confirmed for small e in 2D.

Lower bounds established for relative error.

Exact values computed for three and four terminals.

Abstract

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.