节点不相交多径生成器及其与容错生成器的关系
Node-Disjoint Multipath Spanners and their Relationship with Fault-Tolerant Spanners
摘要：
受多路径路由的启发，我们引入了一个多连接的sparkle变体。为此，我们引入了$p$ -两个节点$u$和$v$之间的多路径成本作为$u$和$v$之间的$p$内部顶点不相交路径的集合的最小权重。给定一个加权图$G$，如果对于所有$u,v$，$H$中的$u$和$v$之间的$p$-多路径成本至多是$s$乘以$G$中的$p$ -多路径成本，则子图$H$是$p$ -多路径$s$ -。$s$因子称为拉伸。最近的容错spectrum的结果的基础上，我们展示了如何建立$p$ -多路径spectrum的恒定拉伸和$\tO(n^{1+1/k})$边缘，固定参数$p$和$k$，$n$是图的节点数。这样的spatial可以通过在$O(k)$轮中运行的分布式算法来构造。此外，我们给出了一个改进的情况下$p=k=2$的建设。我们的$H$具有$O(n^{3/2})$边，并且任意两个节点之间的$H$中的$p$ -多路径成本至多是$G$中的相应成本加上$O(W)$的两倍，$W$是最大边权重。
英文摘要：
Motivated by multipath routing, we introduce a multi-connected variant of spanners. For that purpose we introduce the $p$-multipath cost between two nodes $u$ and $v$ as the minimum weight of a collection of $p$ internally vertex-disjoint paths between $u$ and $v$. Given a weighted graph $G$, a subgraph $H$ is a $p$-multipath $s$-spanner if for all $u,v$, the $p$-multipath cost between $u$ and $v$ in $H$ is at most $s$ times the $p$-multipath cost in $G$. The $s$ factor is called the stretch. Building upon recent results on fault-tolerant spanners, we show how to build $p$-multipath spanners of constant stretch and of $\tO(n^{1+1/k})$ edges, for fixed parameters $p$ and $k$, $n$ being the number of nodes of the graph. Such spanners can be constructed by a distributed algorithm running in $O(k)$ rounds. Additionally, we give an improved construction for the case $p=k=2$. Our spanner $H$ has $O(n^{3/2})$ edges and the $p$-multipath cost in $H$ between any two node is at most twice the corresponding one in $G$ plus $O(W)$, $W$ being the maximum edge weight.
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