Low-Distortion Embeddings of Trees

Abstract

We prove that every tree T=(V,E) on n vertices with edges of unit length can be embedded in the plane with distortion O(√n); that is, we construct a mapping f V→R² such that ρ(u,v) ≤ ||f(u)−f(v)|| ≤ O(√n)·ρ(u,v) for every u,v ∈ V, where ρ(u,v) denotes the length of the path from u to v in T. The embedding is described by a simple and easily computable formula. This is asymptotically optimal in the worst case. We also construct interesting optimal embeddings for a special class of trees (fans consisting of paths of the same length glued together at a common vertex).