Let G=(V,E) be an n -vertices m -edges directed graph with edge weights in the range [1, W ] for some parameter W , and sϵ V be a designated source. In this article, we address several variants of the problem of maintaining the (1+ε)-approximate shortest path from s to each v ϵ V { s } in the presence of a failure of an edge or a vertex. From the graph theory perspective, we show that G has a subgraph H with Õ(ε -1 } n log W ) edges such that for any x,vϵ V , the graph H \ x contains a path whose length is a (1+ε)-approximation of the length of the shortest path from s to v in G \ x . We show that the size of the subgraph H is optimal (up to logarithmic factors) by proving a lower bound of Ω (ε -1 n log W ) edges. Demetrescu, Thorup, Chowdhury, and Ramachandran (SICOMP 2008) showed that the size of a fault tolerant exact shortest path subgraph in weighted directed/undirected graphs is Ω ( m ). Parter and Peleg (ESA 2013) showed that even in the restricted case of unweighted undirected graphs, the size of any subgraph for the exact shortest path is at least Ω ( n 1.5 ). Therefore, a (1+ε)-approximation is the best one can hope for. We consider also the data structure problem and show that there exists an ϕ(ε -1 n log W ) size oracle that for any vϵ V reports a (1+ε)-approximate distance of v from s on a failure of any xϵ V in O(log log 1+ε ( nW )) time. We show that the size of the oracle is optimal (up to logarithmic factors) by proving a lower bound of Ω (ε -1 n log W log -1 n ). Finally, we present two distributed algorithms . We present a single-source routing scheme that can route on a (1+ε)-approximation of the shortest path from a fixed source s to any destination t in the presence of a fault. Each vertex has a label and a routing table of ϕ(ε -1 log W ) bits. We present also a labeling scheme that assigns each vertex a label of ϕ(ε -1 log W ) bits. For any two vertices x,vϵ V , the labeling scheme outputs a (1+ε)-approximation of the distance from s to v in G \ x using only the labels of x and v .

中文翻译：

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近似单源容错最短路径

让G=(V,E)豆角，扁豆n-顶点米-edges 有向图，边权重在 [1,W] 对于某些参数W， 和εV成为指定来源。在本文中，我们解决了维持 (1+ε) 近似最短路径问题的几个变体s每个vε五{s} 在边或顶点出现故障的情况下。来自图论观点，我们表明G有一个子图H与 Õ(ε-1}n日志W) 边缘使得对于任何x,vϵ V, 图H\x包含一条路径，其长度是从s到v在G\x. 我们证明了子图的大小H通过证明 Ω (ε-1 n日志W) 边缘。Demetrescu、Thorup、Chowdhury 和 Ramachandran (SICOMP 2008) 表明，加权有向/无向图中容错精确最短路径子图的大小为 Ω (米）。Parter 和 Peleg (ESA 2013) 表明，即使在未加权无向图的受限情况下，确切最短路径的任何子图的大小至少为 Ω (n 1.5）。因此，(1+ε)-近似是最好的。我们还考虑数据结构问题并证明存在一个 φ(ε-1 n日志W) 大小的预言机，对于任何ⅤⅤⅤ报告 (1+ε) - 近似距离v从s在任何失败xϵ V在 O(log 日志1+ε(净重）） 时间。我们通过证明 Ω (ε-1 n日志W日志-1 n）。最后，我们介绍两个分布式算法. 我们提供单一来源路由方案可以在固定源的最短路径的 (1+ε) 近似上进行路由s到任何目的地吨在存在故障的情况下。每个顶点都有一个标签和一个 φ(ε-1日志W) 位。我们还提出了一个标签方案为每个顶点分配一个 φ(ε-1日志W) 位。对于任意两个顶点x,vϵ V，标记方案输出距离的 (1+ε)-近似值s到v在 G \X使用只要的标签X和v.