Shortest paths problems are subject to extensive studies in classic distributed models such as the CONGEST or congested clique, which describe the way in which nodes may communicate in order to solve such a problem. %where nodes initially know only the distance to their neighbors in some graph and must compute distances to any other node with as few communication rounds as possible. This article focuses on hybrid networks, which give nodes access to multiple, different modes of communication, in particular the HYBRID model which combines unrestricted local communication along edges of the input graph alongside heavily restricted global communication between arbitrary pairs of nodes. Previous work [Augustine et al, SODA'20, Kuhn et al. PODC'20] showed that each node learning its distance to $k$ dedicated source nodes (aka the $k$-SSP problem) takes at least $\tilOm(\!\sqrt{k})$ rounds in the HYBRID model, even for polynomial approximations. This lower bound was matched with algorithmic solutions for $k \geq n^{2/3}$. However, as $k$ gets smaller, the gap between the known upper and lower bounds diverges and even becomes exponential for the single source shortest paths problem (SSSP). In this work we plug this gap for the whole range of $k$ (up to terms that are polylogarithmic in $n$), by giving algorithmic solutions for $k$-SSP in $\tilO\big(\!\sqrt k\big)$ rounds for any $k$.

翻译：最短路径问题在经典分布式模型（如CONGEST或拥塞团）中受到广泛研究，这些模型描述了节点为求解此类问题而进行通信的方式。本文聚焦于混合网络，该网络允许节点使用多种不同的通信模式，特别是HYBRID模型——该模型结合了沿输入图边的无限制局部通信，以及任意节点对间高度受限的全局通信。先前工作[Augustine等, SODA'20; Kuhn等, PODC'20]表明，即使对于多项式近似，每个节点学习其到$k$个指定源节点的距离（即$k$-SSP问题）至少需要$\tilOm(\!\sqrt{k})$轮。该下界与$k \geq n^{2/3}$的算法解决方案匹配。然而，随着$k$减小，已知上下界之间的差距逐渐扩大，对于单源最短路径问题（SSSP）甚至呈指数级。本文通过为任意$k$的$k$-SSP问题提供$\tilO\big(\!\sqrt k\big)$轮的算法解决方案，填补了$k$整个取值范围（直至$n$的多对数项）的这一差距。