稀疏容错BFS树
Sparse Fault-Tolerant BFS Trees
摘要：
本文讨论了设计一棵稀疏的{ \em fault-tolerant} BFS树（简称{ \em FT-BFS树}）的问题，即给定网络$G$的稀疏子图$T$，使得在单个边或顶点失效后，$T$的幸存部分$T'$仍然包含$G$（的幸存部分）的BFS生成树.我们的主要结果如下。我们提出了一个算法，对于每个$n$ -顶点图$G$和源节点$s$构建一个（单边故障）FT-BFS树植根于$s$与$O(n \cdot \min\{\Depth(s), \sqrt{n}\})$边，其中$\Depth(s)$是BFS树植根于$s$的深度。这一结果是由一个匹配的下界补充，表明存在$n$ -顶点图与源节点$s$，任何边（或顶点）FT-BFS树植根于$s$有$Ω(n^{3/2})$的边缘。然后，我们考虑{ \em容错多源BFS树}，或简称{ \em FT-MBFS树}，旨在为源$S\subseteq V$的某个子集提供（故障后）以每个源$s\in S$为根的BFS树。再次，提供了紧边界，表明存在多时间算法，对于每个$n$顶点图和大小为$σ$的源集$S \subseteq V$，（单故障）FT-MBFS树$T^*(S)$从每个源$s_i \in S$，具有$O(\sqrtσ \cdot n^{3/2})$边，并且另一方面存在具有基数$σ$的源集合$S \subseteq V$的$n$ -顶点图，其上来自$S$的任何FT-MBFS树具有$Ω(\sqrtσ\cdot n^{3/2})$边。最后，我们提出了一个构造FT-BFS和FT-MBFS结构的$O(\log n)$近似算法。后者是补充了硬度的结果，说明不存在$Ω(\log n)$近似算法，这些问题在标准的复杂性假设。
英文摘要：
This paper addresses the problem of designing a sparse {\em fault-tolerant} BFS tree, or {\em FT-BFS tree} for short, namely, a sparse subgraph $T$ of the given network $G$ such that subsequent to the failure of a single edge or vertex, the surviving part $T'$ of $T$ still contains a BFS spanning tree for (the surviving part of) $G$. Our main results are as follows. We present an algorithm that for every $n$-vertex graph $G$ and source node $s$ constructs a (single edge failure) FT-BFS tree rooted at $s$ with $O(n \cdot \min\{\Depth(s), \sqrt{n}\})$ edges, where $\Depth(s)$ is the depth of the BFS tree rooted at $s$. This result is complemented by a matching lower bound, showing that there exist $n$-vertex graphs with a source node $s$ for which any edge (or vertex) FT-BFS tree rooted at $s$ has $Ω(n^{3/2})$ edges. We then consider {\em fault-tolerant multi-source BFS trees}, or {\em FT-MBFS trees} for short, aiming to provide (following a failure) a BFS tree rooted at each source $s\in S$ for some subset of sources $S\subseteq V$. Again, tight bounds are provided, showing that there exists a poly-time algorithm that for every $n$-vertex graph and source set $S \subseteq V$ of size $σ$ constructs a (single failure) FT-MBFS tree $T^*(S)$ from each source $s_i \in S$, with $O(\sqrtσ \cdot n^{3/2})$ edges, and on the other hand there exist $n$-vertex graphs with source sets $S \subseteq V$ of cardinality $σ$, on which any FT-MBFS tree from $S$ has $Ω(\sqrtσ\cdot n^{3/2})$ edges. Finally, we propose an $O(\log n)$ approximation algorithm for constructing FT-BFS and FT-MBFS structures. The latter is complemented by a hardness result stating that there exists no $Ω(\log n)$ approximation algorithm for these problems under standard complexity assumptions.
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