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The Steiner tree problem is one of the fundamental and classical problems in combinatorial optimization.In this paper we study this problem in the CONGEST ED CLIQUE model of distributed computing and present two deterministic distributed approximation algorithms for the same. The first algorithm computes a Steiner tree in Õ(n 1/3 ) rounds and Õ(n 7/3 ) messages for a given connected undirected weighted graph of n nodes. Note here that Õ(•) notation hides polylogarithmic factors in n. The second one computes a Steiner tree in O(S + log log n) rounds and O(S (n − t) 2 + n 2 ) messages, where S and t are the shortest path diameter and the number of terminal nodes respectively in the given input graph. Both the algorithms admit an approximation factor of 2(1 − 1/ ), where is the number of terminal leaf nodes in the optimal Steiner tree. For graphs with S = ω(n 1/3 log n), the first algorithm exhibits better performance than the second one in terms of the round complexity. On the other hand, for graphs with S = õ(n 1/3 ), the second algorithm outperforms the first one in terms of the round complexity. In fact when S = O(log log n) then the second algorithm admits a round complexity of O(log log n) and message complexity of Õ(n 2 ). To the best of our knowledge, this is the first work to study the Steiner tree problem in the CONGEST ED CLIQUE model.

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The Steiner tree problem is one of the fundamental and classical problems in combinatorial optimization. In this paper we study this problem in the CONGESTED CLIQUE model (CCM) [29] of distributed computing. For the Steiner tree problem in the CCM, we consider that each vertex of the input graph is uniquely mapped to a processor and edges are naturally mapped to the links between the corresponding processors. Regarding output, each processor should know whether the vertex assigned to it is in the solution or not and which of its incident edges are in the solution. We present two deterministic distributed approximation algorithms for the Steiner tree problem in the CCM. The first algorithm computes a Steiner tree using [Formula: see text] rounds and [Formula: see text] messages for a given connected undirected weighted graph of [Formula: see text] nodes. Note here that [Formula: see text] notation hides polylogarithmic factors in [Formula: see text]. The second one computes a Steiner tree using [Formula: see text] rounds and [Formula: see text] messages, where [Formula: see text] and [Formula: see text] are the shortest path diameter and number of edges respectively in the given input graph. Both the algorithms achieve an approximation ratio of [Formula: see text], where [Formula: see text] is the number of leaf nodes in the optimal Steiner tree. For graphs with [Formula: see text], the first algorithm exhibits better performance than the second one in terms of the round complexity. On the other hand, for graphs with [Formula: see text], the second algorithm outperforms the first one in terms of the round complexity. In fact when [Formula: see text] then the second algorithm achieves a round complexity of [Formula: see text] and message complexity of [Formula: see text]. To the best of our knowledge, this is the first work to study the Steiner tree problem in the CCM.

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