Distributed Approximation of Minimum Routing Cost Trees

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 + 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) stated in [13]. 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 in [13]. In addition we derive a deterministic 2-approximation.