# Distributed Approximation Algorithms for Steiner Tree in the $\mathcal{CONGESTED}$ $\mathcal{CLIQUE}$

## TL;DR

This paper introduces two deterministic distributed approximation algorithms for the Steiner tree problem in the congested clique model, achieving different efficiencies based on graph parameters, and is the first to address this problem in this setting.

## Contribution

The paper presents the first study of Steiner tree approximation algorithms specifically designed for the congested clique model in distributed computing.

## Key findings

- First algorithm runs in O(n^{1/3}) rounds with O(n^{7/3}) messages.
- Second algorithm runs in O(S + \u007f ext{log} ext{ ext{log}} n) rounds with O(S (n - t)^2 + n^2) messages.
- Both algorithms achieve an approximation factor of 2(1 - 1/) based on terminal leaf nodes.

## Abstract

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 $S = \omega(n^{1/3} \log n)$, 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.