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 [MATH HERE] for a graph of n nodes and t terminals, where S is the shortest path diameter of the graph. Note here that the Õ (·) 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 [MATH HERE] of the optimal where ℓ is the number of terminal leaf nodes in the optimal Steiner tree. The round complexity of our algorithm is [MATH HERE] and the message complexity is O(Δ(n − t)S + n3/2, where Δ 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(Δ(n - t)S) respectively.