Massively-parallel graph algorithms have received extensive attention over the past decade, with research focusing on three memory regimes: the superlinear regime, the near-linear regime, and the sublinear regime. The sublinear regime is the most desirable in practice, but conditional hardness results point towards its limitations. In this work we study a \emph{heterogeneous} model, where the memory of the machines varies in size. We focus mostly on the heterogeneous setting created by adding a single near-linear machine to the sublinear MPC regime, and show that even a single large machine suffices to circumvent most of the conditional hardness results for the sublinear regime: for graphs with $n$ vertices and $m$ edges, we give (a) an MST algorithm that runs in $O(\log\log(m/n))$ rounds; (b) an algorithm that constructs an $O(k)$-spanner of size $O(n^{1+1/k})$ in $O(1)$ rounds; and (c) a maximal-matching algorithm that runs in $O(\sqrt{\log(m/n)}\log\log(m/n))$ rounds. We also observe that the best known near-linear MPC algorithms for several other graph problems which are conjectured to be hard in the sublinear regime (minimum cut, maximal independent set, and vertex coloring) can easily be transformed to work in the heterogeneous MPC model with a single near-linear machine, while retaining their original round complexity in the near-linear regime. If the large machine is allowed to have \emph{superlinear} memory, all of the problems above can be solved in $O(1)$ rounds.

翻译：大规模并行图算法在过去十年中受到了广泛关注，研究聚焦于三种内存模式：超线性模式、近线性模式和亚线性模式。亚线性模式在实践中最为理想，但条件性困难结果揭示了其局限性。本文研究了一种**异构**模型，其中各机器的内存大小各异。我们主要关注通过在亚线性MPC环境中添加一台近线性机器所产生的异构场景，并证明即使单台大容量机器也足以规避亚线性模式中大多数条件性困难结果：对于具有$n$个顶点和$m$条边的图，我们提出（a）一种运行时间为$O(\log\log(m/n))$轮的最小生成树算法；（b）一种在$O(1)$轮内构造大小为$O(n^{1+1/k})$的$O(k)$-稀疏子图算法；以及（c）一种运行时间为$O(\sqrt{\log(m/n)}\log\log(m/n))$轮的最大匹配算法。我们还观察到，其他几个被推测在亚线性模式下难以解决的图问题（最小割、最大独立集和顶点着色）的已知最优近线性MPC算法，可以轻松转换到带有单台近线性机器的异构MPC模型中运行，同时保留其在近线性模式下的原始轮复杂度。若允许大容量机器拥有**超线性**内存，上述所有问题均可在$O(1)$轮内解决。