Distributed Approximation of Minimum Routing Cost Trees

TL;DR

This paper presents new distributed algorithms for approximating the Minimum Routing Cost Spanning Tree problem, achieving better approximation ratios and runtimes in the message passing model, especially for unweighted graphs.

Contribution

It introduces a randomized $(2+ ext{epsilon})$-approximation with improved runtime and a deterministic 2-approximation for the problem in distributed settings.

Findings

Achieved a $(2+ ext{epsilon})$-approximation with runtime $O(D+rac{ ext{log} n}{ ext{epsilon}})$.

Improved previous algorithms with $O( ext{log} n)$ approximation and $O(D ext{log}^2 n)$ runtime.

Derived an optimal $O(D)$ runtime for $O( ext{log} n)$-approximations.

Abstract

We study the NP-hard problem of approximating a Minimum Routing Cost Spanning Tree in the message passing model with limited bandwidth (CONGEST model). In this problem one tries to find a spanning tree of a graph $G$ over $n$ nodes that minimizes the sum of distances between all pairs of nodes. In the considered model every node can transmit a different (but short) message to each of its neighbors in each synchronous round. We provide a randomized $(2+\epsilon)$-approximation with runtime $O(D+\frac{\log n}{\epsilon})$ for unweighted graphs. Here, $D$ is the diameter of $G$. This improves over both, the (expected) approximation factor $O(\log n)$ and the runtime $O(D\log^2 n)$ of the best previously known algorithm. Due to stating our results in a very general way, we also derive an (optimal) runtime of $O(D)$ when considering $O(\log n)$-approximations as done by the best previously known algorithm. In addition we derive a deterministic $2$-approximation.