Improved distributed approximation for Steiner tree in the CONGEST model

Abstract

In this paper we present two deterministic distributed algorithms for the Steiner tree (ST) problem in the CONGEST model. The first algorithm computes a 2(1−1/ℓ)-approximate ST using O(S+nlog⁎⁡n) rounds and O(mS+n3/2) messages for a graph of n nodes and m edges, where S is the shortest path diameter of the graph and ℓ is the number of leaf nodes in the optimal ST. It improves the round complexity of the best distributed ST algorithm known so far, which is O˜(S+min{St,n})[34], where t is the number of terminal nodes. The second algorithm improves the message complexity of the first one by dropping the additive term of O(n3/2) at the expense of a logarithmic multiplicative factor in the round complexity. We also show that for graphs with S=O(log⁡n), a 2(1−1/ℓ)-approximate ST can be deterministically computed using O˜(n) rounds and O˜(m) messages and these complexities almost coincide with the results of some of the singularly-optimal minimum spanning tree (MST) algorithms proposed in [15,22,37].