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 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(n2) time and space algorithm that computes a best swap edge of every tree edge. Previously, an O(n2 log4 n) time and O(n2 + mlog2 n) 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(n2) time and space algorithm would be considered a breakthrough.

A novel algorithm for the all-best-swap-edge problem on tree spanners

Bilò Davide;
2018

Abstract

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 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(n2) time and space algorithm that computes a best swap edge of every tree edge. Previously, an O(n2 log4 n) time and O(n2 + mlog2 n) 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(n2) time and space algorithm would be considered a breakthrough.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11697/179737
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? ND
social impact