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，这反映了现代通信网络的非均匀特性。混合模型的先前工作主要关注存在性最优算法，即存在一个病态实例族，使得任何算法在其上都无法表现更好。但这忽略了实践中此类最坏情况实例通常不会出现或可被主动规避的事实。本文聚焦于由Garay、Kutten和Peleg（FOCS 1993）首次提出的通用最优性概念。粗略而言，通用最优算法是指：给定任意输入图，其运行速度与专门为该图设计的最佳算法相同。我们展示了混合模型中首批通用最优算法。针对混合模型中广播和多消息单播等基础信息传播任务，我们提出了通用最优解决方案。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，获得了通用最优方案。本文的一个主要概念性贡献是提出了名为邻域质量的新图参数，它刻画了混合模型中许多基础图问题的内在复杂度。我们还展示了混合模型中新的存在性最优最短路径算法，这些算法作为关键子程序被用于我们的通用最优算法，并具有独立的研究价值。针对$k$源最短路径问题的新算法对所有$k$值都匹配了原有的$\tilde{\Omega}(\sqrt{k})$下界。此前，该下界仅在$k \in \tilde{\Omega}(n^{2/3})$时才被证明是紧致的。