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arXiv 2108.01740cs.DC
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混合模型中的确定性分布式算法及其下界

Deterministic Distributed Algorithms and Lower Bounds in the Hybrid Model

Ioannis Anagnostides, Themis Gouleakis

摘要:

$\hybrid$模型最近由Augustine等人\cite{DBLP:conf/soda/AugustineHKSS20}引入,以便从算法的观点来表征组合多种通信模式的网络的能力。具体地说,它是假设,标准的$\local$模型的分布式计算增强了所有的所有通信的功能,但非常有限的带宽,由节点容量限制集团($\ncc$)捕获。在这项工作中,我们提供了一些新的见解的权力,混合网络的分布式算法的基本问题。首先,我们提出了一个确定性算法,解决任何问题的稀疏$n$节点图在$\widetilde{\mathcal{O}}(\sqrt{n})$轮的$\hybrid$。我们结合这个原语与几个稀疏化技术,以获得有效的分布式算法一般图。最值得注意的是,对于所有对最短路径问题,我们给出了确定性$(1 + ε)$ -和$\log n/\log \log n$ -近似算法分别为未加权和加权图与轮复杂度$\widetilde{\mathcal{O}}(\sqrt{n})$在$\hybrid$,密切匹配的性能的最先进的随机算法的库恩和施耐德\cite{10.1145/3382734.3405719}。此外,我们与低拥塞捷径\cite{DBLP:conf/soda/GhaffariH16}的Ghaffari-Haeupler框架建立了联系,导致-除其他外-在$\log^{\mathcal{O}(1)}n$轮之后,即使我们限制局部边缘以每轮传输$\mathcal{O}(\log n)$ -位,也有很高的概率,最小切割的$(1 + ε)$ -近似算法。最后,我们证明了通过减少从集合不相交问题,$\widetildeΩ(n^{1/3})$轮需要确定半径的未加权图,以及$(3/2 - ε)$ -近似加权图。

英文摘要:

The $\hybrid$ model was recently introduced by Augustine et al. \cite{DBLP:conf/soda/AugustineHKSS20} in order to characterize from an algorithmic standpoint the capabilities of networks which combine multiple communication modes. Concretely, it is assumed that the standard $\local$ model of distributed computing is enhanced with the feature of all-to-all communication, but with very limited bandwidth, captured by the node-capacitated clique ($\ncc$). In this work we provide several new insights on the power of hybrid networks for fundamental problems in distributed algorithms. First, we present a deterministic algorithm which solves any problem on a sparse $n$-node graph in $\widetilde{\mathcal{O}}(\sqrt{n})$ rounds of $\hybrid$. We combine this primitive with several sparsification techniques to obtain efficient distributed algorithms for general graphs. Most notably, for the all-pairs shortest paths problem we give deterministic $(1 + ε)$- and $\log n/\log \log n$-approximate algorithms for unweighted and weighted graphs respectively with round complexity $\widetilde{\mathcal{O}}(\sqrt{n})$ in $\hybrid$, closely matching the performance of the state of the art randomized algorithm of Kuhn and Schneider \cite{10.1145/3382734.3405719}. Moreover, we make a connection with the Ghaffari-Haeupler framework of low-congestion shortcuts \cite{DBLP:conf/soda/GhaffariH16}, leading -- among others -- to a $(1 + ε)$-approximate algorithm for Min-Cut after $\log^{\mathcal{O}(1)}n$ rounds, with high probability, even if we restrict local edges to transfer $\mathcal{O}(\log n)$-bits per round. Finally, we prove via a reduction from the set disjointness problem that $\widetildeΩ(n^{1/3})$ rounds are required to determine the radius of an unweighted graph, as well as a $(3/2 - ε)$-approximation for weighted graphs.

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