We consider the problem of computing compact routing tables for a (weighted) planar graph $G:= (V, E,w)$ in the PRAM, CONGEST, and the novel HYBRID communication model. We present algorithms with polylogarithmic work and communication that are almost optimal in all relevant parameters, i.e., computation time, table sizes, and stretch. All algorithms are heavily randomized, and all our bounds hold w.h.p. For a given parameter $\epsilon>0$, our scheme computes labels of size $\widetilde{O}(\epsilon^{-1})$ and is computed in $\widetilde{O}(\epsilon^{-2})$ time and $\widetilde{O}(n)$ work in the PRAM and the HYBRID model and $\widetilde{O}(\epsilon^{-2} \cdot HD)$ (Here, $HD$ denotes the network's hop-diameter) time in CONGEST. The stretch of the resulting routing scheme is $1+\epsilon$. To achieve these results, we extend the divide-and-conquer framework of Li and Parter [STOC '19] and combine it with state-of-the-art distributed distance approximation algorithms [STOC '22]. Furthermore, we provide a distributed decomposition scheme, which may be of independent interest.

翻译：我们研究在PRAM、CONGEST以及新型HYBRID通信模型下，为（加权）平面图 $G:= (V, E,w)$ 计算紧凑路由表的问题。我们提出了具有多对数级别工作量和通信量的算法，这些算法在所有相关参数（即计算时间、路由表大小和伸展度）上均达到近乎最优。所有算法均高度随机化，且所有界在高概率下成立。对于给定参数 $\epsilon>0$，我们的方案生成大小为 $\widetilde{O}(\epsilon^{-1})$ 的标签，在PRAM和HYBRID模型中以 $\widetilde{O}(\epsilon^{-2})$ 时间及 $\widetilde{O}(n)$ 工作量完成计算，在CONGEST模型中以 $\widetilde{O}(\epsilon^{-2} \cdot HD)$（此处$HD$表示网络的跳数直径）时间完成计算。所得路由方案的伸展度为 $1+\epsilon$。为实现这些结果，我们扩展了Li和Parter [STOC '19] 的分治框架，并将其与最先进的分布式距离近似算法 [STOC '22] 相结合。此外，我们提出了一种可能具有独立价值的分布式分解方案。