Network decomposition is a central concept in the study of distributed graph algorithms. We present the first polylogarithmic-round deterministic distributed algorithm with small messages that constructs a strong-diameter network decomposition with polylogarithmic parameters. Concretely, a ($C$, $D$) strong-diameter network decomposition is a partitioning of the nodes of the graph into disjoint clusters, colored with $C$ colors, such that neighboring clusters have different colors and the subgraph induced by each cluster has a diameter at most $D$. In the weak-diameter variant, the requirement is relaxed by measuring the diameter of each cluster in the original graph, instead of the subgraph induced by the cluster. A recent breakthrough of Rozho\v{n} and Ghaffari [STOC 2020] presented the first $\text{poly}(\log n)$-round deterministic algorithm for constructing a weak-diameter network decomposition where $C$ and $D$ are both in $\text{poly}(\log n)$. Their algorithm uses small $O(\log n)$-bit messages. One can transform their algorithm to a strong-diameter network decomposition algorithm with similar parameters. However, that comes at the expense of requiring unbounded messages. The key remaining qualitative question in the study of network decompositions was whether one can achieve a similar result for strong-diameter network decompositions using small messages. We resolve this question by presenting a novel technique that can transform any black-box weak-diameter network decomposition algorithm to a strong-diameter one, using small messages and with only moderate loss in the parameters.

中文翻译：

---

强直径网络分解

网络分解是分布式图形算法研究的中心概念。我们提出了第一个带有小消息的多对数舍入确定性分布式算法，该算法构造了具有多对数参数的强直径网络分解。具体来说，（$ C $，$ D $）强直径网络分解是将图的节点划分为不相交的簇，并用$ C $颜色进行着色，以使相邻簇具有不同的颜色，并且每个子图都诱发子图簇的直径最大为$ D $。在弱直径变量中，通过测量原始图中每个簇的直径（而不是由簇引起的子图）来放宽要求。Rozho \ v {n}和Ghaffari的最新突破[STOC 2020]提出了第一个$ \ text {poly}（\ log n）$舍入确定性算法，用于构造弱直径网络分解，其中$ C $和$ D $都在$ \ text {poly}（\ log n）$中。他们的算法使用小的$ O（\ log n）$位消息。可以将他们的算法转换为具有相似参数的大直径网络分解算法。但是，这是以需要无限制的消息为代价的。在网络分解研究中，剩下的关键定性问题是，是否可以使用小消息对强直径网络分解实现类似的结果。通过提出一种可以将任何黑匣子弱直径网络分解算法转换为强直径算法的新颖技术，我们解决了这一问题，