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 : 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. Jünger;

Class representative computation using graph embedding and clustering

“…Noktalar arasındaki uzaklık ise dügümler arasındaki ilişkiyi göstermektedir. Çizge gömme ile ilgili daha ayrıntılı bilgi için [9], [1]'e bakılabilir.…”

Plane embeddings of planar graph metrics

“…Roughly speaking, the goal is to embed a given metric (matrix of pairwise distances among n points) into a target space while minimizing the maximum additive or multiplicative error, called distortion, introduced in the distances. 1 Of particular interest in many of these applications is embedding into low-dimensional geometric spaces, typically Euclidean. For example, in visualization, the natural target spaces are 2D and 3D Euclidean space, for display on an LCD panel or a holographic display.…”

“…Even when the target space is the one-dimensional line, little is known. For example, when the given metric is the shortest-path metric of an (unweighted) tree, the best known approximation factor for multiplicative distortion isÕ(n 1/3 ) (improving on the O(n 1/2 )-approximation for general graphs) [5], and it is unknown whether it is possible to achieve a factor of n o (1) ; see [3]. (On the other hand, additive distortion is less interesting in this context: there is an O(1)-approximation for embedding a general metric into the line [12].…”

“…For additive distortion and the Euclidean plane, the only 1 More precisely, given a metric M on a point set V and a target space of d-dimensional s space, the goal is to find a map- o(n)-approximation algorithm runs in pseudo-quasipolynomial time [4]. For additive distortion and the 1 plane, there is a polynomial-time O(1)-approximation [2].…”

“…Motivated by applications in visualization, ad-hoc networks, and molecular reconstruction, we consider the natural problem of embedding shortest-path metrics of unweighted planar graphs (planar graph metrics) into the Euclidean plane. It is known that, in the special case of shortest-path metrics of trees, embedding into the plane requires Θ( √ n) distortion in the worst case [19,1], and surprisingly, this worst-case upper bound provides the best known approximation algorithm for minimizing distortion. We answer an open question posed in this work and highlighted by Matoušek [21] by proving that some planar graph metrics require Ω(n 2/3 ) distortion in any embedding into the plane, proving the first separation between these two types of graph metrics.…”