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 ofÕ(n 1+1/k ) edges 1 , 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.

Finding single-source shortest $p$-disjoint paths: fast computation and sparse preservers

“…A p-multipath α-spanner of G is a spanning subgraph H of G containing, for each pair of vertices u, v, p edge-disjoint paths from u to v of total cost at most α times the cost of the cheapest p edge-disjoint paths from u to v in G. Among other results, the authors of [17] prove the existence, for any choice of p, of a p-multipath p(2k − 1)-spanner of size O(pn 1+ 1 k ) for undirected graphs. Following [17], there has been further work on multipath spanners [11,18]. All of the above papers, however, focus on approximated costs, in the all-pairs setting on undirected graphs.…”

Multipath Spanners via Fault-Tolerant Spanners

“…This can be done by invoking a construction for s-spanner, in the algorithm for constructing vertex fault-tolerant spanners of Dinitz and Krauthgamer in [9], that satisfies the following property, defined in [16]. Every edge e is either included in the spanner H or H contains an alternative path to e of length at most s · ω(e) and with at most s hops.…”

“…In [16] . In this paper, we significantly improve the stretch bounds from the two previous papers, using the related concept of fault-tolerant s-spanners, introduced in [6] for general graphs.…”

“…This paper significantly improves the stretch-size tradeoff result of the two previous papers, using the related concept of fault-tolerant s-spanners, introduced in [6] for general graphs. More precisely, we show that at the cost of increasing the number of edges by a polynomial factor in p and s, it is possible to obtain an s-multipath spanner, thereby improving on the large stretch obtained in [15,16]. …”

“…An s-spanner H of a graph G is a subgraph such that the distance between any two vertices u and v in H is greater by at most a multiplicative factor s than the distance in G. In this paper, we focus on an extension of the concept of spanners to p-multipath distance, defined as the smallest length of a collection of p pairwise (vertex or edge) disjoint paths. The notion of multipath spanners was introduced in [15,16] for edge (respectively, vertex) disjoint paths. This paper significantly improves the stretch-size tradeoff result of the two previous papers, using the related concept of fault-tolerant s-spanners, introduced in [6] for general graphs.…”