  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.
