连接二元关系的近最优并行算法
A Near-Optimal Parallel Algorithm for Joining Binary Relations
AI总结：
提出MPC模型下常数轮连接二元关系的近最优算法，负载匹配下界，核心为孤立笛卡尔积定理，并最优解决子图枚举问题。
摘要：
我们提出了在大规模并行计算（MPC）模型中评估每个输入关系具有两个属性的自然连接的常数轮算法。我们的算法实现了负载 $\ ilde{O}(m/p^{1/ρ})$，其中 $m$ 是输入关系的总大小，$p$ 是机器数量，$ρ$ 是连接的分数边覆盖数，$\ ilde{O}(.)$ 隐藏了多对数因子。该负载与已知下界匹配，相差仅一个多对数因子。所提出算法的核心是一个新定理（我们称之为“孤立笛卡尔积定理”），它为问题的数学结构提供了新的见解。我们的结果意味着子图枚举问题（目标是报告常数大小子图模式的所有出现）可以在 MPC 模型中以最优方式（相差一个多对数因子）解决。
英文摘要：
We present a constant-round algorithm in the massively parallel computation (MPC) model for evaluating a natural join where every input relation has two attributes. Our algorithm achieves a load of $\tilde{O}(m/p^{1/ρ})$ where $m$ is the total size of the input relations, $p$ is the number of machines, $ρ$ is the join's fractional edge covering number, and $\tilde{O}(.)$ hides a polylogarithmic factor. The load matches a known lower bound up to a polylogarithmic factor. At the core of the proposed algorithm is a new theorem (which we name the "isolated cartesian product theorem") that provides fresh insight into the problem's mathematical structure. Our result implies that the subgraph enumeration problem, where the goal is to report all the occurrences of a constant-sized subgraph pattern, can be settled optimally (up to a polylogarithmic factor) in the MPC model.
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