Greedy Embeddings, Trees, and Euclidean vs. Lobachevsky Geometry

A greedy embedding of an unweighted undirected graph G = (V, E) into a metric space (X, ρ) is a function f : V → X such that for every source-sink pair of different vertices s, t ∈ V it is the case that s has a neighbor v in G with ρ(f(v), f(t)) < ρ(f(s), f(t)). Finding greedy embeddings of connectivity graphs helps to build distributed routing schemes with compact routing tables. In this paper we take a refined look at greedy embeddings, previously addressed in [1, 2], by examining their description complexity as a key parameter in conjunction with their dimensionality. We give arguments showing that the dimensionality lower-bounds for monotone maps do not extend to greedy embeddings. We prove a unified O(log n) lower-bound on the dimension of no-stretch greedy embeddings when the host metric is Euclidean or Lobachevsky geometry. The essence of the lower bound entails showing that low-dimensional spaces lack the topological capacity to realize the embeddings of certain graphs with “hard crossroads.” This technique might be of independent interest. We develop new methods for building concise embeddings of trees (and some other graphs) in 3-dimensional Lobachevsky spaces using recursive applications of hyperbolic isometries guided by caterpillar-like decompositions. Our embeddings improve over prior work [1] by achieving O(κ(T ) · log n) description complexity, where κ(T ) is the caterpillar dimension. We further demonstrate concise O(log n)-dimensional greedy embeddings of trees into Euclidean space using techniques inspired by [3], thereby strengthening our belief and intuition that all graphs can be embedded with no stretch in ` n) 2 .