无需嵌入的低拥塞快捷方式
Low-Congestion Shortcuts without Embedding
摘要：
分布式优化算法经常面临着解决网络中不相交的连接部分上的子问题。不幸的是，这些部分的直径可能远远大于底层网络的直径，导致运行时间缓慢。最近的工作[Ghaffari和Hauepler; SODA'16]表明，这种现象可以被看作是广泛的根本原因的普遍$Ω(\sqrt{n} + D)$下限，适用于大多数优化问题的CONGEST模型。从积极的方面来看，这项工作还引入了低拥塞捷径，作为在某些感兴趣的拓扑中规避此问题的优雅解决方案。特别是，他们证明了任何平面网络都存在很好的捷径，更一般地说，任何有界亏格网络都存在。这直接导致快速$O(D \log^{O(1)} n)$分布式算法MST和最小割近似，给定的人可以有效地构建这些快捷方式在一个分布式的方式。不幸的是，[Ghaffari和Hauepler; SODA'16]的快捷构造在很大程度上依赖于网络的亏格嵌入。然而，分布式地计算这样的嵌入是一个困难的问题-即使是平面网络。有界亏格图的分布式嵌入算法还没有见到。在这项工作中，我们侧步这个问题，通过定义一个限制和更结构化的形式的捷径，并给出了一种新的建设算法，有效地找到一个捷径，这是一个对数因子，作为最好的捷径，存在于一个给定的网络。这种新的构造算法直接导致了一个$O(D \log^{O(1)} n)$轮算法，用于解决优化问题，如MST的任何拓扑结构，其中存在良好的限制快捷方式-而不需要计算任何嵌入。这包括有界亏格图的第一个有效算法。
英文摘要：
Distributed optimization algorithms are frequently faced with solving sub-problems on disjoint connected parts of a network. Unfortunately, the diameter of these parts can be significantly larger than the diameter of the underlying network, leading to slow running times. Recent work by [Ghaffari and Hauepler; SODA'16] showed that this phenomenon can be seen as the broad underlying reason for the pervasive $Ω(\sqrt{n} + D)$ lower bounds that apply to most optimization problems in the CONGEST model. On the positive side, this work also introduced low-congestion shortcuts as an elegant solution to circumvent this problem in certain topologies of interest. Particularly, they showed that there exist good shortcuts for any planar network and more generally any bounded genus network. This directly leads to fast $O(D \log^{O(1)} n)$ distributed algorithms for MST and Min-Cut approximation, given that one can efficiently construct these shortcuts in a distributed manner. Unfortunately, the shortcut construction of [Ghaffari and Hauepler; SODA'16] relies heavily on having access to a genus embedding of the network. Computing such an embedding distributedly, however, is a hard problem - even for planar networks. No distributed embedding algorithm for bounded genus graphs is in sight. In this work, we side-step this problem by defining a restricted and more structured form of shortcuts and giving a novel construction algorithm which efficiently finds a shortcut which is, up to a logarithmic factor, as good as the best shortcut that exists for a given network. This new construction algorithm directly leads to an $O(D \log^{O(1)} n)$-round algorithm for solving optimization problems like MST for any topology for which good restricted shortcuts exist - without the need to compute any embedding. This includes the first efficient algorithm for bounded genus graphs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
