Distributed Approximation Algorithms for Steiner Tree in the $\mathcal{CONGESTED}$ $\mathcal{CLIQUE}$
The \emph{Steiner tree} problem is one of the fundamental and classical problems in combinatorial optimization. In this paper, we study this problem in the $\mathcal{CONGESTED}$ $\mathcal{CLIQUE}$ model of distributed computing and present two deterministic distributed approximation algorithms for the same. The first algorithm computes a Steiner tree in $\tilde{O}(n^{1/3})$ rounds and $\tilde{O}(n^{7/3})$ messages for a given connected undirected weighted graph of $n$ nodes. Note here that $\tilde{O}(\cdot)$ 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 \emph{shortest path diameter} and the number of \emph{terminal} nodes respectively in the given input graph. Both the algorithms admit an approximation factor of $2(1 - 1/\ell)$, where $\ell$ 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 $S = \tilde{o}(n^{1/3})$, 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 $\tilde{O}(n^2)$. To the best of our knowledge, this is the first work to study the Steiner tree problem in the $\mathcal{CONGESTED}$ $\mathcal{CLIQUE}$ model.