Low-Distortion Embeddings of Trees

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). Communicated by: P. Mutzel and M. J¨unger; submitted May 2002; revised April 2003. The research was supported by project LN00A056 of the Ministry of Education of the Czech Republic and by Charles University grants No. 158/99 and 159/99. 

• Type: Conference
• Year: 2001
• Where: GD
• Authors: Robert Babilon, Jirí Matousek, Jana Maxová, Pavel Valtr
• Keywords: Computer Graphics, GD 2001, Interesting Optimal Embeddings, Mapping F, Unit Length