Fully Scalable Massively Parallel Algorithms for Embedded Planar Graphs
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We consider the 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 Tětek [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 𝒮 = n^2/3 + Ω(1). In this work, we provide an improved recursive framework to obtain constant-round algorithms in the more challenging fully scalable regime where memory size per machine can be 𝒮 = n^δ for any given constant δ > 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 ε-emulators can be incorporated into our recursive framework to obtain constant-round (1+ε)-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+ε)-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+ε)-approximate all-pairs shortest paths (APSP), diameter, and radius.
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