This paper addresses the problem of designing a β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-additive fault-tolerant approximate BFS (or FT-ABFS for short) structure, namely, a subgraph H of the network G such that subsequent to the failure of a single edge e, the surviving part of H still contains an approximate BFS spanning tree for (the surviving part of) G, whose distances satisfy dist(s,v,H\{e})≤dist(s,v,G\{e})+β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{dist}(s,v,H{\setminus } \{e\}) \le \mathrm{dist}(s,v,G{\setminus } \{e\})+\beta $$\end{document} for every v∈V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v \in V$$\end{document}. It was shown in Parter and Peleg (SODA, 2014), that for every β∈[1,O(logn)]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in [1, O(\log n)]$$\end{document} there exists an n-vertex graph G with a source s for which any β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-additive FT-ABFS structure rooted at s has Ω(n1+ϵ(β))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (n^{1+\epsilon (\beta )})$$\end{document} edges, for some function ϵ(β)∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon (\beta ) \in (0,1)$$\end{document}. In particular, 3-additive FT-ABFS structures admit a lower bound of Ω(n5/4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (n^{5/4})$$\end{document} edges. In this paper we present the first upper bound, showing that there exists a poly-time algorithm that for every n-vertex unweighted undirected graph G and source s constructs a 4-additive FT-ABFS structure rooted at s with at most O(n4/3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n^{4/3})$$\end{document} edges. The main technical contribution of our algorithm is in adapting the path-buying strategy used in Baswana et al. (ACM Trans Algorithms 7:A5, 2010) and Cygan et al. (Proceedings of the 30th symposium on theoretical aspects of computer science, pp 209–220, 2013) to failure-prone settings.

中文翻译：

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具有附加拉伸的容错近似 BFS 结构

G\{e})+β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek } \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{dist}(s,v,H{\setminus } \{e\}) \le \mathrm{dist}(s,v ,G{\setminus } \{e\})+\beta $$\end{document} 对于每个 v∈V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v \in V$$\end{document}。在 Parter 和 Peleg (SODA, 2014) 中表明，对于每个 β∈[1，1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin }{-69pt} \begin{document}$$\epsilon (\beta ) \in (0,1)$$\end{document}。特别是，3-additive FT-ABFS 结构允许下界 Ω(n5/4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \ usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (n^{5/4})$$\end{document} 边. 在本文中，我们提出了第一个上限，表明存在一种多时间算法，该算法对于每个 n 顶点未加权无向图 G 和源 s 构造一个以 s 为根的 4-additive FT-ABFS 结构，最多为 O(n4/3)\documentclass[12pt]{minimal } \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document }$$O(n^{4/3})$$\end{document} 边。我们算法的主要技术贡献是调整了 Baswana 等人使用的路径购买策略。(ACM Trans Algorithms 7:A5, 2010) 和 Cygan 等人。（第 30 届计算机科学理论方面研讨会论文集，第 209-220 页，2013 年）到容易失败的设置。