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. This phenomenon can be seen as the broad underlying reason for the pervasive Ω~(n+D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Omega }(\sqrt{n} + D)$$\end{document} lower bounds that apply to most optimization problems in the CONGEST model. On the positive side, [Ghaffari and Hauepler; SODA’16] 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(DlogO(1)n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(D \log ^{O(1)} n)$$\end{document} 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 tree-restricted shortcuts: a more structured and restricted form of shortcuts. We give a novel construction algorithm which efficiently finds such shortcuts that are, up to a logarithmic factor, as good as the best restricted shortcuts that exist for a given network. This new construction algorithm directly leads to an O(DlogO(1)n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(D \log ^{O(1)} n)$$\end{document}-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 greatly simplifies the existing planar algorithms and includes the first efficient algorithm for bounded genus graphs.

中文翻译：

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无需嵌入的低拥塞快捷方式

分布式优化算法经常面临解决网络不相交连接部分上的子问题。不幸的是，这些部分的直径可能明显大于底层网络的直径，从而导致运行时间变慢。这种现象可以看作是普遍存在的 Ω~(n+D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage 的广泛根本原因{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Omega }(\sqrt{n} + D)$$\end{文档}下限适用于 CONGEST 模型中的大多数优化问题。从积极的方面来说，[Ghaffari 和 Hauepler; SODA'16] 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，他们表明任何平面网络都存在良好的捷径，更普遍的是任何有界属网络。这直接导致快速 O(DlogO(1)n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs } \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(D \log ^{O(1)} n)$$\end{document} MST 和 Min 分布式算法-Cut 近似，因为可以以分布式方式有效地构建这些捷径。不幸的是，[Ghaffari 和 Hauepler; SODA'16] 在很大程度上依赖于访问网络的类嵌入。然而，分布式计算这样的嵌入是一个难题——即使对于平面网络也是如此。看不到有界属图的分布式嵌入算法。在这项工作中，我们通过定义树限制的快捷方式来回避这个问题：一种更加结构化和受限的快捷方式。我们给出了一种新颖的构造算法，它可以有效地找到这样的捷径，这些捷径在对数因子内与给定网络存在的最佳受限捷径一样好。这种新的构造算法直接导致 O(DlogO(1)n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \ usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(D \log ^{O(1)} n)$$\end{document}-round算法用于为任何存在良好受限捷径的拓扑解决优化问题，如 MST——无需计算任何嵌入。这大大简化了现有的平面算法，并包括第一个有效的有界属图算法。