Distributed Algorithms for Low Stretch Spanning Trees

Ruben Becker, Yuval Emek, Mohsen Ghaffari, Christoph Lenzen, Michael Wagner
2019 International Symposium on Distributed Computing  
Given an undirected graph with integer edge lengths, we study the problem of approximating the distances in the graph by a spanning tree based on the notion of stretch. Our main contribution is a distributed algorithm in the CONGEST model of computation that constructs a random spanning tree with the guarantee that the expected stretch of every edge is O(log 3 n), where n is the number of nodes in the graph. If the graph is unweighted, then this algorithm can be implemented to run in O(D)
more » ... , where D is the hop-diameter of the graph, thus being asymptotically optimal. In the weighted case, the run-time of our algorithm matches the currently best known bound for exact distance computations, i.e.,Õ(min{ √ nD, √ nD 1/4 + n 3/5 + D}). We stress that this is the first distributed construction of spanning trees leading to poly-logarithmic expected stretch with non-trivial running time. solution depends on how well T preserves the relevant structure of G. Such transformations have been a key driver in many of the algorithmic developments in the past two decades, in centralized approximation algorithms. Our focus in this paper is on distance-related graph problems and transformations of graphs into spanning trees that approximately preserve distances, based on the notion of stretch. Spanning Trees and Stretch is the restriction of to the edges in E T . By definition, the distances in T are at least as large as those in G, namely, d T (u, v) ≥ d G (u, v) for every two vertices u, v ∈ V , where the distances d T (·, ·) and d G (·, ·) are defined with respect to the edge length functions T and , respectively. The notion of stretch provides a bound in the converse direction: given an edge e = (u, v) ∈ G, the stretch of e in T is defined to be str T (e) = d T (u, v)/ (e).
doi:10.4230/lipics.disc.2019.4 dblp:conf/wdag/BeckerEGL19 fatcat:lpbwemj3fvfwfc4vpnbshlyoqi