低记忆MPC模型中的快速规则集算法
Sample-and-Gather: Fast Ruling Set Algorithms in the Low-Memory MPC Model
摘要：
受对称性破缺问题（如低内存大规模并行计算（MPC）模型中的最大独立集（MIS）和最大匹配）的最新进展的启发（例如，Behnezhad等人~ PODC 2019; Ghaffari-Uitto SODA 2019），我们研究了该模型中规则集问题的复杂性。MPC模型作为大规模分布式计算的模型已经变得非常流行，并且它具有每台机器的内存在输入大小上是强次线性的约束。对于图问题，已经设计了极快的MPC算法，假设每机器存储器为$\tildeΩ(n)$，其中$\tildeΩ(n)$是图中的节点数（例如，Ghaffari等人的$O(\log\log n)$ MIS算法，PODC 2018）。然而，已经证明在低内存MPC模型中为图形问题设计快速MPC算法要困难得多，其中每台机器的内存被限制为在节点数量上是强次线性的，即，$O(n^\eps)$用于$0 < \eps < 1$。在本文中，我们提出了一个算法的2-规则集问题，运行在$\tilde{O}(\log^{1/6} Δ)$轮whp，在低内存MPC模型。然后，我们将这一结果推广到$β$统治集的任何整数$β> 1$。具体来说，我们表明可以在低内存MPC模型中计算$β$规则集，其中每台机器的$O(n^\eps)$内存在$\tilde{O}(β\cdot \log^{1/(2^{β+1}-2)} Δ)$轮中，whp。由此可以立即得出，用于$β= Ω(\log\log\log Δ)$ -规则集的$β$-规则集可以仅在$O(β\log\log n)$轮whp中计算。上述结果假设总存储器为$\tilde{O}(m + n^{1+\eps})$。我们还提出了算法$β$ -统治集在低内存MPC模型假设所有机器的总内存被限制为$\tilde{O}(m)$。
英文摘要：
Motivated by recent progress on symmetry breaking problems such as maximal independent set (MIS) and maximal matching in the low-memory Massively Parallel Computation (MPC) model (e.g., Behnezhad et al.~PODC 2019; Ghaffari-Uitto SODA 2019), we investigate the complexity of ruling set problems in this model. The MPC model has become very popular as a model for large-scale distributed computing and it comes with the constraint that the memory-per-machine is strongly sublinear in the input size. For graph problems, extremely fast MPC algorithms have been designed assuming $\tildeΩ(n)$ memory-per-machine, where $n$ is the number of nodes in the graph (e.g., the $O(\log\log n)$ MIS algorithm of Ghaffari et al., PODC 2018). However, it has proven much more difficult to design fast MPC algorithms for graph problems in the low-memory MPC model, where the memory-per-machine is restricted to being strongly sublinear in the number of nodes, i.e., $O(n^\eps)$ for $0 < \eps < 1$. In this paper, we present an algorithm for the 2-ruling set problem, running in $\tilde{O}(\log^{1/6} Δ)$ rounds whp, in the low-memory MPC model. We then extend this result to $β$-ruling sets for any integer $β> 1$. Specifically, we show that a $β$-ruling set can be computed in the low-memory MPC model with $O(n^\eps)$ memory-per-machine in $\tilde{O}(β\cdot \log^{1/(2^{β+1}-2)} Δ)$ rounds, whp. From this it immediately follows that a $β$-ruling set for $β= Ω(\log\log\log Δ)$-ruling set can be computed in in just $O(β\log\log n)$ rounds whp. The above results assume a total memory of $\tilde{O}(m + n^{1+\eps})$. We also present algorithms for $β$-ruling sets in the low-memory MPC model assuming that the total memory over all machines is restricted to $\tilde{O}(m)$.
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