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.

中文翻译：

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拥挤集团中斯坦纳树的分布式逼近算法

施泰纳树问题是组合优化中的基础和经典问题之一。在本文中，我们在分布式计算的拥挤集团模型 (CCM) [29] 中研究了这个问题。对于 CCM 中的 Steiner 树问题，我们认为输入图的每个顶点都唯一地映射到一个处理器，并且边自然地映射到相应处理器之间的链接。关于输出，每个处理器都应该知道分配给它的顶点是否在解中，以及它的哪些入射边在解中。我们针对 CCM 中的 Steiner 树问题提出了两种确定性分布式逼近算法。第一个算法使用 [Formula: see text] 轮次和 [Formula: see text] 消息为给定的 [Formula: 见文本]节点。请注意，[公式：参见文本] 表示法在 [公式：参见文本] 中隐藏了多对数因子。第二个使用 [Formula: see text] rounds 和 [Formula: see text] 消息计算 Steiner 树，其中 [Formula: see text] 和 [Formula: see text] 分别是最短路径直径和边数给定的输入图。两种算法都达到了[公式：见文本]的近似比，其中[公式：见文本]是最优施泰纳树中叶节点的数量。对于具有 [公式：参见文本] 的图，第一种算法在轮复杂度方面表现出比第二种算法更好的性能。另一方面，对于具有 [公式：参见文本] 的图，第二个算法在轮复杂度方面优于第一个算法。实际上当 [公式：见文本]，则第二个算法达到[公式：见文本]的轮复杂度和[公式：见文本]的消息复杂度。据我们所知，这是在 CCM 中研究 Steiner 树问题的第一项工作。