In recent years, a number of applications of ad-hoc net

We introduce a novel measure called ε-four-points condition (ε-4PC), which assigns a value ε ∈ [0, 1] to every metric space quantifying how close the metric is to a tree metric. Data-sets taken from real Internet measurements indicate remarkable closeness of Internet latencies to tree metrics based on this condition. We study embeddings of ε-4PC metric spaces into trees and prove tight upper and lower bounds. Specifically, we show that there are constants c1 and c 2 such that, (1) every metric (X, d) which satisfies the ε-4PC can be embedded into a tree with distortion (1 + ε) c 1 log |X| , and (2) for every ε ∈ [0, 1] and any number of nodes, there is a metric space (X, d) satisfying the ε-4PC that does not embed into a tree with distortion less than (1 + ε) c 2 log |X| . In addition, we prove a lower bound on approximate distance labelings of ε-4PC metrics, and give tight bounds for tree embeddings with additive error guarantees.

Existing empirical studies of Internet structure and path properties indicate that the Internet is tree-like. This work quantifies the degree to which at least two important Internet measures-latency and bandwidth-approximate tree metrics. We evaluate our ability to model end-to-end measures using tree embeddings by actually building tree representations. In addition to being simple and intuitive models, these trees provide a range of commonly-required functionality beyond serving as an analytical tool.The contributions of our study are twofold. First, we investigate the ability to portray the inherent hierarchical structure of the Internet using the most pure and compact topology, trees. Second, we evaluate the ability of our compact representation to facilitate many natural tasks, such as the selection of servers with short latency or high bandwidth from a client. Experiments show that these tasks can be done with high degree of success and modest overhead.

Abstract