Fully Scalable Massively Parallel Algorithms for Embedded Planar Graphs

We consider the \emph{massively parallel computation} (MPC) model, which is a theoretical abstraction of large-scale parallel processing models such as MapReduce. In this model, assuming the widely believed 1-vs-2-cycles conjecture, it is not possible to solve many basic graph problems in constant rounds with strongly sublinear memory size per machine. Recently, Holm and T\v{e}tek [SODA 2023] showed that it is possible to get around this barrier for planar graphs when a planar straight-line embedding of the graph is given. For such inputs on $n$ vertices, they obtained constant-round MPC algorithms for connected components, minimum spanning tree (MST), and $O(1)$-approximation of $st$-shortest path, diameter, and radius, as long as the memory size per machine is $\mathcal{S} = n^{2/3 + \Omega(1)}$. In this work, we provide an improved recursive framework to obtain constant-round algorithms in the more challenging \emph{fully scalable} regime where memory size per machine can be $\mathcal{S} = n^\delta$ for any given constant $\delta > 0$. This gives the first constant-round algorithms in this regime for fundamental problems such as connected components, MST, and EMST. Moreover, we show that $\varepsilon$-emulators can be incorporated into our recursive framework to obtain constant-round $(1+\varepsilon)$-approximation algorithms for single source shortest path (SSSP) and shortest cycle in embedded planar graphs. We show that it is possible to construct a dual graph of the given embedded planar graph in constant rounds, which allows us to solve the $(1+\varepsilon)$-approximate $st$-maximum flow and minimum cut problem as both reduce to a shortest cycle problem in the dual graph. Using $O(n^2)$ total space, we also obtain constant-round algorithms for $(1+\varepsilon)$-approximate all-pairs shortest paths (APSP), diameter, and radius.