Near-Optimal Compact Routing and Decomposition Schemes for Planar Graphs

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 a 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.