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

TL;DR

This paper introduces p-multipath spanners, a new type of graph substructure that approximates fault-tolerant routing paths with constant stretch and efficient construction, extending previous fault-tolerant spanner results.

Contribution

It defines p-multipath spanners and provides methods to construct them with constant stretch and near-linear edges, including an improved case for p=k=2.

Findings

Constructed p-multipath spanners with constant stretch and O(n^{1+1/k}) edges.

Developed a distributed algorithm for spanner construction in O(k) rounds.

Achieved an improved construction for p=k=2 with O(n^{3/2}) edges and a 2-approximate p-multipath cost.

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, 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.