A linear-time approximation algorithm for the minimum-length geometric embedding of trees

Abstract

In this paper, we study the problem of computing a geometric embedding of a tree onto a point set in the Euclidean plane such that the total edge length of the embedding is minimum. We present a linear-time O ( Δph ( T ) ) -approximation algorithm, where Δ and ph ( T ) denote the maximum node-degree and the path height of the input tree T, respectively. The previous best result in the literature has O ( nlog ⁡ log ⁡ n ) time-complexity and O ( Δ log 2 ⁡ n ) approximation factor. We show that ph ( T ) is less than log 2 ⁡ n for any n-node tree T. The problem is a generalization of the Euclidean travelling salesman problem, which is a well-known NP-hard problem.