A faster computation of all the best swap edges of a tree spanner

Given a 2-edge connected, positively real-weighted graph G with n vertices and m edges, a tree σ-spanner of G is a spanning tree T in which for every pair of vertices, the ratio of their distance in T over that in G is bounded by σ, the so-called stretch factor of T. Tree spanners with provably good stretch factors find applications in communication networks, distributed systems, and network design, but unfortunately –as any tree-based infrastructure– they are highly sensitive to even a single link failure, since this results in a network disconnection. Thus, when such an event occurs, the overall effort that has to be afforded to rebuild an effective tree spanner (i.e., computational costs, set-up of new links, updating of the routing tables, etc.) can be prohibitive. However, if the edge failure is only transient, these costs can simply be avoided, by promptly reestablishing the connectivity through a careful selection of a temporary swap edge, i.e., an edge in G reconnecting the two subtrees of T induced by the edge failure. According to the tree spanner’s nature, a best swap edge for a failing edge e is then a swap edge generating a reconnected tree of minimum stretch factor w.r.t. distances in the graph G deprived of edge e. For this problem we provide two efficient linear-space solutions for both the weighted and the unweighted case, running in O(m2 logα(m,n)) and O(mn logn) time, respectively. As discussed in the paper, our algorithms also improve on the time complexity of previous results provided for other related settings of the problem.