A Novel Algorithm for the All-Best-Swap-Edge Problem on Tree Spanners
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Given a 2-edge connected undirected graph G with n vertices and m edges, a σ-tree spanner is a spanning tree 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 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 best swap edge, i.e. an edge that reconnects T-e at a minimum stretch factor. More precisely, we design an O(n^2) time and space algorithm that computes a best swap edge of every tree edge. Previously, an O(n^2 ^4 n) time and O(n^2+m^2n) space algorithm was known for edge-weighted graphs (Bilò 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(n^2) time and space algorithm would be considered a breakthrough.
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