Round-Message Trade-Off In Distributed Steiner Tree Construction In The Congest Model

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 O(S + root n log* n) rounds whose cost is optimal upto a factor of 2(1 - 1/l), where n and S are the number of nodes and shortest path diameter [17] respectively of the given input graph and l is the number of terminal leaf nodes in the optimal Steiner tree. The message complexity of the algorithm is O(Delta(n - t)S + n(3/2)), which is equivalent to O(mS + n(3/2)), where. is the maximum degree of a vertex in the graph, t is the number of terminal nodes (we assume that t < n), 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 <(O)over tilde> (S + root n) rounds and (O) over tilde (mS) messages and still achieves an approximation factor of 2(1 - 1/l). Note here that (O) over tilde (center dot) notation hides polylogarithmic factors in n. This algorithm improves the message complexity of Saikia and Karmakar's algorithm by dropping the additive term of O(n(3/2)) 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 (S = O(log n)), a 2(1- 1/l)-approximate Steiner tree can be computed in (O) over tilde (root n) rounds and (O) over tilde (m) 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].