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, solving many basic graph problems in $O(1)$ rounds with a strongly sublinear memory size per machine is impossible. We improve on the recent work of Holm and T\v{e}tek [SODA 2023] that bypass this barrier for problems when a planar embedding of the graph is given. In the previous work, on graphs of size $n$ with $O(n/\mathcal{S})$ machines, the memory size per machine needs to be at least $\mathcal{S} = n^{2/3+\Omega(1)}$, whereas we extend their work to the fully scalable regime, where the memory size per machine can be $\mathcal{S} = n^{\delta}$ for any constant $0< \delta < 1$. We give the first constant round fully scalable algorithms for embedded planar graphs for the problems of (i) connectivity and (ii) minimum spanning tree (MST). Moreover, we show that the $\varepsilon$-emulator of Chang, Krauthgamer, and Tan [STOC 2022] can be incorporated into our recursive framework to obtain constant-round $(1+\varepsilon)$-approximation algorithms for the problems of computing (iii) single source shortest path (SSSP), (iv) global min-cut, and (v) $st$-max flow. All previous results on cuts and flows required linear memory in the MPC model. Furthermore, our results give new algorithms for problems that implicitly involve embedded planar graphs. We give as corollaries constant round fully scalable algorithms for (vi) 2D Euclidean MST using $O(n)$ total memory and (vii) $(1+\varepsilon)$-approximate weighted edit distance using $\widetilde{O}(n^{2-\delta})$ memory. Our main technique is a recursive framework combined with novel graph drawing algorithms to compute smaller embedded planar graphs in constant rounds in the fully scalable setting.

翻译：我们考虑大规模并行计算（MPC）模型，该模型是对MapReduce等大规模并行处理模型的理论抽象。在该模型中，假设广泛接受的1-vs-2-cycles猜想成立，则在每台机器内存大小强次线性条件下，用$O(1)$轮解决许多基本图问题是不可能的。我们改进了Holm和T\v{e}tek [SODA 2023]的最新工作，该工作针对给定平面嵌入的图绕过这一障碍。先前工作中，对于规模为$n$、使用$O(n/\mathcal{S})$台机器的图，每台机器内存大小至少需要$\mathcal{S} = n^{2/3+\Omega(1)}$，而我们将他们的工作扩展到完全可扩展情形，其中每台机器内存大小可为任意常数$0< \delta < 1$下的$\mathcal{S} = n^{\delta}$。我们首次提出了嵌入平面图在以下问题上的常数轮完全可扩展算法：(i) 连通性 以及 (ii) 最小生成树（MST）。此外，我们证明Chang、Krauthgamer和Tan [STOC 2022]的$\varepsilon$-仿真器可融入我们的递归框架，从而获得以下问题的常数轮$(1+\varepsilon)$-近似算法：(iii) 单源最短路径（SSSP），(iv) 全局最小割，以及 (v) $st$-最大流。所有此前关于割和流的结果在MPC模型中都需要线性内存。进一步地，我们的结果为隐式涉及嵌入平面图的问题提供了新算法。作为推论，我们给出了以下常数轮完全可扩展算法：(vi) 使用$O(n)$总内存的二维欧几里得最小生成树，以及 (vii) 使用$\widetilde{O}(n^{2-\delta})$内存的$(1+\varepsilon)$-近似加权编辑距离。我们的主要技术是一个递归框架，结合新颖的图绘制算法，在完全可扩展设置下于常数轮内计算更小的嵌入平面图。