In theoretical computer science, it is a common practice to show existential lower bounds for problems, meaning there is a family of pathological inputs on which no algorithm can do better. However, most inputs of interest can be solved much more efficiently, giving rise to the notion of universally optimal algorithms, which run as fast as possible on every input. Questions on the existence of universally optimal algorithms were first raised by Garay, Kutten, and Peleg in FOCS '93. This research direction reemerged recently through a series of works, including the influential work of Haeupler, Wajc, and Zuzic in STOC '21, which resolves some of these decades-old questions in the supported CONGEST model. We work in the HYBRID distributed model, which analyzes networks combining both global and local communication. Much attention has recently been devoted to solving distance related problems, such as All-Pairs Shortest Paths (APSP) in HYBRID, culminating in a $\tilde \Theta(n^{1/2})$ round algorithm for exact APSP. However, by definition, every problem in HYBRID is solvable in $D$ (diameter) rounds, showing that it is far from universally optimal. We show the first universally optimal algorithms in HYBRID, by presenting a fundamental tool that solves any broadcasting problem in a universally optimal number of rounds, deterministically. Specifically, we consider the problem in a graph $G$ where a set of $k$ messages $M$ distributed arbitrarily across $G$, requires every node to learn all of $M$. We show a universal lower bound and a matching, deterministic upper bound, for any graph $G$, any value $k$, and any distribution of $M$ across $G$. This broadcasting tool opens a new exciting direction of research into showing universally optimal algorithms in HYBRID. As an example, we use it to obtain algorithms for approximate and exact APSP in general and sparse graphs.

翻译：在理论计算机科学中，常通过证明问题的存在性下界来表明存在一类病态输入，使得任何算法都无法表现更优。然而，大多数实际感兴趣的输入可通过更高效的算法求解，由此引出通用最优算法的概念——这类算法能在每个输入上实现尽可能快的运行速度。关于通用最优算法存在性的问题最早由Garay、Kutten和Peleg于FOCS '93提出。这一研究方向近期通过一系列工作重新兴起，包括Haeupler、Wajc和Zuzic在STOC '21提出的具有影响力的研究，该工作解决了支持CONGEST模型中一些由来已久的问题。我们研究的HYBRID分布式模型分析了同时包含全局通信与局部通信的网络。近年来，距离相关问题的求解（如HYBRID中的全源最短路径问题）备受关注，最新成果提出了精确APSP的$\tilde \Theta(n^{1/2})$轮算法。然而，根据定义，HYBRID中所有问题均可通过$D$（直径）轮求解，这表明该模型远未达到通用最优。我们通过提出一种基础工具，在HYBRID模型中首次展示了通用最优算法——该工具能在任意广播问题上以确定性方式实现通用最优轮数。具体而言，我们考虑图$G$中的问题：$k$条消息$M$任意分布于$G$中，要求每个节点学习所有$M$。对于任意图$G$、任意$k$值以及$M$在$G$中的任意分布，我们证明了通用下界及其匹配的确定性上界。该广播工具为在HYBRID模型中证明通用最优算法开辟了令人振奋的新研究方向。作为示例，我们利用该工具在一般图与稀疏图中获得了近似与精确APSP的算法。