Multipath Spanners via Fault-Tolerant Spanners

Abstract

Abstract. 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. 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].

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.…”

“…Since a direct computation of π t would be too time-consuming, the idea is that of exploiting the suboptimality property of Lemma 4 to consider π t as the composition of two subpaths π q and π t [q : t], where q = q(π t ) and π q is a shortest path from s to q in G q i−1 . To this aim, we follow a Dijkstra-like approach (see lines [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. More precisely, once we have computed π q , we attempt to extend it towards every other vertex t by concatenating π q with a shortest path from q to t in G t i−1 .…”