Online, Dynamic, and Distributed Embeddings of Approximate Ultrametrics

The theoretical computer science community has traditionally used embeddings of finite metrics as a tool in designing approximation algorithms. Recently, however, there has been considerable interest in using metric embeddings in the context of networks to allow network nodes to have more knowledge of the pairwise distances between other nodes in the network. There has also been evidence that natural network metrics like latency and bandwidth have some nice structure, and in particular come close to satisfying an -three point condition or an four point condition. This empirical observation has motivated the study of these special metrics, including strong results about embeddings into trees and ultrametrics. Unfortunately all of the current embeddings require complete knowledge about the network up front, and so are less useful in real networks which change frequently. We give the first metric embeddings which have both low distortion and require only small changes in the structure of the embedding when the network changes. In particular, we give an embedding of semimetrics satisfying an -three point condition into ultrametrics with distortion (1 + ) n+4 and the property that any new node requires only O(n) amortized edge swaps, where we use the number of edge swaps as a measure of “structural change”. This notion of structural change naturally leads to small update messages in a distributed implementation in which every node has a copy of the embedding. The natural offline embedding has only (1+ ) n distortion but can require Ω(n) amortized edge swaps per node addition. This online embedding also leads to a natural dynamic algorithm that can handle node removals as well as insertions.