Given a 2-edge connected, unweighted, and undirected graph G with n
vertices and m edges, a σ-tree spanner is a spanning tree T of G
in which the ratio between the distance in T of any pair of vertices and the
corresponding distance in G is upper bounded by σ. The minimum value
of σ for which T is a σ-tree spanner of G is also called the
{\em stretch factor} of T. We address the fault-tolerant scenario in which
each edge e of a given tree spanner may temporarily fail and has to be
replaced by a {\em best swap edge}, i.e. an edge that reconnects T−e at a
minimum stretch factor. More precisely, we design an O(n2) time and space
algorithm that computes a best swap edge of every tree edge. Previously, an
O(n2log4n) time and O(n2+mlog2n) space algorithm was known for
edge-weighted graphs [Bil\`o et al., ISAAC 2017]. Even if our improvements on
both the time and space complexities are of a polylogarithmic factor, we stress
the fact that the design of a o(n2) time and space algorithm would be
considered a breakthrough.Comment: The paper has been accepted for publication at the 29th International
Symposium on Algorithms and Computation (ISAAC 2018). 12 pages, 3 figure