Minor Excluded Network Families Admit Fast Distributed Algorithms

Bernhard Haeupler, Jason Li, Goran Zuzic
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引用次数: 31

Abstract

Distributed network optimization problems, such as minimum spanning tree, minimum cut, and shortest path, are an active research area in distributed computing. This paper presents a fast distributed algorithm for such problems in the CONGEST model, on networks that exclude a fixed minor. On general graphs, many optimization problems, including the ones mentioned above, require Ω(√ n) rounds of communication in the CONGEST model, even if the network graph has a much smaller diameter. Naturally, the next step in algorithm design is to design efficient algorithms which bypass this lower bound on a restricted class of graphs. Currently, the only known method of doing so uses the low-congestion shortcut framework of Ghaffari and Haeupler [SODA'16]. Building off of their work, this paper proves that excluded minor graphs admit high-quality shortcuts, leading to an Õ(D^2) round algorithm for the aforementioned problems, where D is the diameter of the network graph. To work with excluded minor graph families, we utilize the Graph Structure Theorem of Robertson and Seymour. To the best of our knowledge, this is the first time the Graph Structure Theorem has been used for an algorithmic result in the distributed setting. Even though the proof is involved, merely showing the existence of good shortcuts is sufficient to obtain simple, efficient distributed algorithms. In particular, the shortcut framework can efficiently construct near-optimal shortcuts and then use them to solve the optimization problems. This, combined with the very general family of excluded minor graphs, which includes most other important graph classes, makes this result of significant interest.
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次要排除的网络家庭承认快速分布式算法
最小生成树、最小切割和最短路径等分布式网络优化问题是分布式计算领域的研究热点。本文提出了一种快速的分布式算法来解决这类问题,在CONGEST模型中,网络中排除了一个固定的分支。在一般图上,许多优化问题,包括上面提到的问题,在CONGEST模型中需要Ω(√n)轮通信,即使网络图的直径要小得多。自然,算法设计的下一步是设计有效的算法,绕过限制类图的下界。目前,唯一已知的方法是使用Ghaffari和Haeupler的低拥塞捷径框架[SODA'16]。在他们工作的基础上,本文证明了被排除的小图承认高质量的捷径,从而得到了用于上述问题的Õ(D^2)轮算法,其中D是网络图的直径。为了处理排除的次要图族,我们利用了Robertson和Seymour的图结构定理。据我们所知,这是图结构定理第一次被用于分布式设置的算法结果。尽管涉及到证明,但仅仅表明存在良好的捷径就足以获得简单、高效的分布式算法。特别地,该快捷方式框架可以有效地构造近最优快捷方式,并利用它们来求解优化问题。这一点,再加上排除次要图的非常一般的家族,其中包括大多数其他重要的图类,使得这个结果非常有趣。
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