Round-Message Trade-Off in Distributed Steiner Tree Construction in the CONGEST Model

Abstract

The Steiner tree problem is one of the fundamental optimization problems in distributed graph algorithms. Recently Saikia and Karmakar [27] proposed a deterministic distributed algorithm for the Steiner tree problem that constructs a Steiner tree in (formula presented) rounds whose cost is optimal upto a factor of (formula presented), where n and S are the number of nodes and shortest path diameter [17] respectively of the given input graph and (formula presented) is the number of terminal leaf nodes in the optimal Steiner tree. The message complexity of the algorithm is (formula presented), which is equivalent to (formula presented), where (formula presented) is the maximum degree of a vertex in the graph, t is the number of terminal nodes (we assume that (formula presented)), and m is the number of edges in the given input graph. This algorithm has a better round complexity than the previous best algorithm for Steiner tree construction due to Lenzen and Patt-Shamir [21]. In this paper we present a deterministic distributed algorithm which constructs a Steiner tree in (formula presented) rounds and (formula presented) messages and still achieves an approximation factor of (formula presented). Note here that (formula presented) notation hides polylogarithmic factors in n. This algorithm improves the message complexity of Saikia and Karmakar’s algorithm by dropping the additive term of (formula presented) at the expense of a logarithmic multiplicative factor in the round complexity. Furthermore, we show that for sufficiently small values of the shortest path diameter (formula presented), a (formula presented)-approximate Steiner tree can be computed in (formula presented) rounds and (formula presented) messages and these complexities almost coincide with the results of some of the singularly-optimal minimum spanning tree (MST) algorithms proposed in [9, 12, 23].