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

Abstract

A metric tree embedding of expected stretch α ≥ 1 maps a weighted n -node graph G = ( V , E , ω) to a weighted tree T = ( V T , E T , ω T ) with V ⊑ V T such that, for all v , w ∈ V , dist( v , w , G ) ≤ dist( v , w , T ), and E[dist( v , w , T )] ≤ α dist( v , w , G ). 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 ∈ O(log n ) requires Ω ( n 2 ) work and a metric as input. In this article, we show how to achieve the same guarantees using polylog n depth and Õ( m 1+ε ) work, where m = | E | and ε > 0 is an arbitrarily small constant. Moreover, one may further reduce the work to Õ( m + n 1+ε ) at the expense of increasing the expected stretch to O(ε −1 log n ). 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 variety of previous “Moore-Bellman-Ford-like” algorithms, to be of independent interest and discuss it in depth. In our tree embedding algorithm, we leverage it to provide efficient query access to an approximate metric that allows sampling the tree using polylog n depth and Õ( m ) work. We illustrate the generality and versatility of our techniques by various examples and a number of additional results. Specifically, we (1) improve the state of the art for determining metric tree embeddings in the Congest model, (2) determine a (1 + εˆ)-approximate metric regarding the distances in a graph G in polylogarithmic depth and Õ( n ( m + n 1 + ε )) work, and (3) improve upon the state of the art regarding the k -median and the buy-at-bulk network design problems.