Parallel Metric Tree Embedding based on an Algebraic View on Moore-Bellman-Ford

A metric tree embedding of expected stretch α maps a weighted n-node graph G = (V, E, w) to a weighted tree T = (VT, ET, wT) with V ⊆ VT, and dist(v, w, G) ≤ dist(v, w, T) and E[dist(v, w, T)] ≤ α dist(v, w, G) for all v, w ∈ V. Such embeddings are highly useful for designing fast approximation algorithms, as many hard problems are easy to solve on tree instances. However, to date the best parallel polylog n depth algorithm that achieves an asymptotically optimal expected stretch of α ∈ Ω(log n) uses Ω(n2) work and requires a metric as input. In this paper, we show how to achieve the same guarantees using Ω(m1+ε) work, where $m$ is the number of edges of G and ε >0 is an arbitrarily small constant. Moreover, one may reduce the work further to Ω(m + n1+ε), at the expense of increasing the expected stretch α to Ω(ε-1 log n) using the spanner construction of Baswana and Sen as preprocessing step. Our main tool in deriving these parallel algorithms is an algebraic characterization of a generalization of the classic Moore-Bellman-Ford algorithm. We consider this framework, which subsumes a large variety of previous "Moore-Bellman-Ford-flavored" algorithms, to be of independent interest.