[论文解读] Improved Purely Additive Fault-Tolerant Spanners

Davide Bilò, Fabrizio Grandoni+3 more|arXiv (Cornell University)|Jul 2, 2015

Complexity and Algorithms in Graphs参考文献 11被引用 10

一句话总结

本文提出了无向无权图中纯加法容错汇聚的改进构造方法，将加法拉伸从6、28和38分别减少至4、10和14，同时保持相同的稀疏性。该方法结合了基于聚类的汇聚增强与容错源汇聚扩展，实现了在边故障情况下的大小与拉伸之间的最优权衡。

ABSTRACT

Let $G$ be an unweighted $n$-node undirected graph. A \emph{$β$-additive spanner} of $G$ is a spanning subgraph $H$ of $G$ such that distances in $H$ are stretched at most by an additive term $β$ w.r.t. the corresponding distances in $G$. A natural research goal related with spanners is that of designing \emph{sparse} spanners with \emph{low} stretch. In this paper, we focus on \emph{fault-tolerant} additive spanners, namely additive spanners which are able to preserve their additive stretch even when one edge fails. We are able to improve all known such spanners, in terms of either sparsity or stretch. In particular, we consider the sparsest known spanners with stretch $6$, $28$, and $38$, and reduce the stretch to $4$, $10$, and $14$, respectively (while keeping the same sparsity). Our results are based on two different constructions. On one hand, we show how to augment (by adding a \emph{small} number of edges) a fault-tolerant additive \emph{sourcewise spanner} (that approximately preserves distances only from a given set of source nodes) into one such spanner that preserves all pairwise distances. On the other hand, we show how to augment some known fault-tolerant additive spanners, based on clustering techniques. This way we decrease the additive stretch without any asymptotic increase in their size. We also obtain improved fault-tolerant additive spanners for the case of one vertex failure, and for the case of $f$ edge failures.

研究动机与目标

• 设计在单条边故障下具有最小加法拉伸的稀疏容错汇聚。
• 在不增加渐近大小的前提下，改进现有纯加法容错汇聚，降低拉伸。
• 开发一种通用框架，将源汇聚容错汇聚转换为完整成对容错汇聚。
• 将该方法扩展至处理多条边故障和顶点故障。

提出的方法

• 通过添加少量边来增强容错源汇聚，以在边故障下保持所有成对距离。
• 应用聚类技术于已知的加法汇聚，以在保持大小边界的同时降低拉伸。
• 使用5-乘法边容错汇聚作为骨干，构造大小为O(n^{4/3})的14-加法EFT汇聚。
• 利用具有4-加法拉伸和Õ(n^{7/5})大小的随机聚类汇聚，构造10-加法EFT汇聚。
• 通过边不相交路径论证证明路径的无故障性，以在边故障下界定拉伸。
• 应用基于性质的距离分析（P2, P3），确保在边移除后拉伸边界仍被保持。

实验结果

研究问题

• RQ1 我们能否在不渐近增加大小的前提下，降低现有稀疏容错汇聚的加法拉伸？
• RQ2 是否可能将源汇聚容错汇聚转换为具有最小开销的完整成对容错汇聚？
• RQ3 对于大小为O(n^{4/3})的EFT汇聚，可实现的最小加法拉伸是多少？
• RQ4 聚类构造方法如何被调整以在容错环境中改善拉伸？
• RQ5 相同技术能否扩展至处理多条边或顶点故障？

主要发现

• 构造出大小为O(n^{4/3})的14-加法边容错汇聚，优于此前同尺寸的38-加法汇聚。
• 实现大小为Õ(n^{7/5})的10-加法边容错汇聚，优于此前同尺寸的28-加法汇聚。
• 利用5-乘法EFT汇聚和6-加法汇聚，多项式时间内构造出14-加法EFT汇聚。
• 通过使用具有Õ(n^{7/5})大小的4-加法汇聚的随机构造，获得10-加法EFT汇聚。
• 该方法使最稀疏已知的大小为O(n^{4/3})的汇聚的拉伸从6降低至4。
• 该框架被扩展以处理f条边故障和顶点故障，实现了更优的拉伸边界。