Simple Distributed Algorithms for Approximating Minimum Steiner Trees

Abstract

Given a network G=(V,E), edge weights w(.), and a set of terminals S ⊆ V, the minimum-weight Steiner tree problem is to find a tree in G that spans S with minimum weight. Most provable heuristics treat the network G is a metric; This assumption, in a distributed setting, cannot be easily achieved without a subtle overhead.

We give a simple distributed algorithm based on a minimum spanning tree heuristic that returns a solution whose cost is within a factor of two of the optimal. The algorithm runs in time O(|V|log|V|) on a synchronous network. We also show that another heuristic based on iteratively finding shortest paths gives a Θ(log |V|)-approximation using a novel charging scheme based on low-congestion routing on trees. Both algorithms work for unit-cost and general cost cases. The algorithms also have applications in finding multicast trees in wireless ad hoc networks.