A simple 2(1-1/l) factor distributed approximation algorithm for steiner tree in the CONGEST model.

The Steiner tree problem is a classical and fundamental problem in combinatorial optimization. The best known deterministic distributed algorithm for the Steiner tree problem in the CONGEST model was proposed by Lenzen and Patt-Shamir [25] that constructs a Steiner tree whose cost is optimal upto a factor of 2 and the round complexity is (O) over tilde (S + root min{St, n}) for a graph of n nodes and t terminals, where S is the shortest path diameter of the graph. Note here that the (O) over tilde(center dot) notation hides polylogarithmic factors in n. In this paper we present a simple deterministic distributed algorithm for constructing a Steiner tree in the CONGEST model with an approximation factor 2(1 - 1/l) of the optimal where l is the number of terminal leaf nodes in the optimal Steiner tree. The round complexity of our algorithm is O(S + root n log* n) and the message complexity is O(Delta(n - t)S + n(3/2)), where. is the maximum degree of a vertex in the graph. Our algorithm is based on the computation of a sub-graph called the shortest path forest for which we present a separate deterministic distributed algorithm with round and message complexities of O(S) and O(Delta(n - t)S) respectively.