In this work we consider the HYBRID model of distributed computing, introduced recently by Augustine, Hinnenthal, Kuhn, Scheideler, and Schneider (SODA 2020), where nodes have access to two different communication modes: high-bandwidth local communication along the edges of the graph and low-bandwidth all-to-all communication, capturing the non-uniform nature of modern communication networks. Prior work in HYBRID has focused on showing existentially optimal algorithms, meaning there exists a pathological family of instances on which no algorithm can do better. This neglects the fact that such worst-case instances often do not appear or can be actively avoided in practice. In this work, we focus on the notion of universal optimality, first raised by Garay, Kutten, and Peleg (FOCS 1993). Roughly speaking, a universally optimal algorithm is one that, given any input graph, runs as fast as the best algorithm designed specifically for that graph. We show the first universally optimal algorithms in HYBRID. We present universally optimal solutions for fundamental information dissemination tasks, such as broadcasting and unicasting multiple messages in HYBRID. Furthermore, we apply these tools to obtain universally optimal solutions for various shortest paths problems in HYBRID. A main conceptual contribution of this work is the conception of a new graph parameter called neighborhood quality that captures the inherent complexity of many fundamental graph problems in HYBRID. We also show new existentially optimal shortest paths algorithms in HYBRID, which are utilized as key subroutines in our universally optimal algorithms and are of independent interest. Our new algorithms for $k$-source shortest paths match the existing $\tilde{\Omega}(\sqrt{k})$ lower bound for all $k$. Previously, the lower bound was only known to be tight when $k \in \tilde{\Omega}(n^{2/3})$.

翻译：本文研究最近由Augustine、Hinnenthal、Kuhn、Scheideler和Schneider（SODA 2020）提出的混合分布式计算模型。在该模型中，节点可访问两种不同的通信模式：沿图边的高带宽局部通信，以及低带宽的全对全通信，这体现了现代通信网络的非均匀特性。先前混合模型的研究主要集中于存在性最优算法，即存在一个病态实例族，使得任何算法在该实例上都无法表现更好。然而，这类最坏情况实例在实际中往往不会出现或可主动避免。本文聚焦于由Garay、Kutten和Peleg（FOCS 1993）首次提出的通用最优性概念。直观而言，通用最优算法是指：对于任意给定的输入图，其运行速度与专门为该图设计的最佳算法相当。我们展示了混合模型中首个通用最优算法，提出了针对混合模型中广播和多播等多消息传播基本任务的通用最优解决方案。此外，我们将这些工具应用于混合模型中各类最短路径问题，获得通用最优解决方案。本文的主要概念贡献是提出了新的图参数——邻域质量（neighborhood quality），该参数刻画了混合模型中许多基本图问题的内在复杂度。我们还提出了混合模型中新的存在性最优最短路径算法，这些算法作为我们通用最优算法的关键子程序，且具有独立研究价值。针对$k$源最短路径问题的新算法对所有$k$均匹配已知的$\tilde{\Omega}(\sqrt{k})$下界，而此前仅当$k \in \tilde{\Omega}(n^{2/3})$时下界已知为紧的。