Node-Disjoint Multipath Spanners and their Relationship with Fault-Tolerant Spanners - Archive ouverte HAL Access content directly
Reports Year : 2011

Node-Disjoint Multipath Spanners and their Relationship with Fault-Tolerant Spanners

Abstract

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\footnote{Tilde-$O$ notation is similar to Big-$O$ up to poly-logarithmic factors in $n$.}, 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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Dates and versions

hal-00622915 , version 1 (13-09-2011)
hal-00622915 , version 2 (16-09-2011)

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Cyril Gavoille, Quentin Godfroy, Laurent Viennot. Node-Disjoint Multipath Spanners and their Relationship with Fault-Tolerant Spanners. 2011. ⟨hal-00622915v2⟩
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