Shortest paths problems are subject to extensive studies in classic distributed models such as the CONGEST or Congested Clique. These models dictate how nodes may communicate in order to determine shortest paths in a distributed input graph. This article focuses on shortest paths problems in the HYBRID model, which combines local communication along edges of the input graph with global communication between arbitrary pairs of nodes that is restricted in terms of bandwidth. Previous work showed that it takes $\tilde \Omega(\!\sqrt{k})$ rounds in the \hybrid model for each node to learn its distance to $k$ dedicated source nodes (aka the $k$-SSP problem), even for crude approximations. This lower bound was also 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 a single source. In this work we close this gap for the whole range of $k$ (up to terms that are polylogarithmic in $n$), by giving algorithmic solutions for $k$-SSP in $\tilde O\big(\!\sqrt k\big)$ rounds for any $k$.

翻译：最短路径问题在经典分布式模型（如 CONGEST 或 Congested Clique）中受到广泛研究。这些模型规定了节点如何通信以确定分布式输入图中的最短路径。本文聚焦于 HYBRID 模型中的最短路径问题，该模型结合了沿输入图边进行的局部通信与受带宽限制的任意节点对之间的全局通信。先前研究表明，在混合模型中，每个节点需要 $\tilde \Omega(\!\sqrt{k})$ 轮才能获得到达 $k$ 个专用源节点的距离（即 $k$-SSP 问题），即使对于粗略近似也是如此。这一下界也与 $k \geq n^{2/3}$ 时的算法解决方案相匹配。然而，随着 $k$ 变小，已知上下界之间的差距逐渐拉大，甚至对于单源情况呈指数增长。本文通过为任意 $k$ 给出 $\tilde O\big(\!\sqrt k\big)$ 轮内求解 $k$-SSP 的算法方案，填补了 $k$ 整个取值范围（至多 $n$ 的多对数项）上的这一差距。