An Improved Algorithm for Computing All the Best Swap Edges of a Tree Spanner

Authors Davide Bilò, Feliciano Colella, Luciano Gualà, Stefano Leucci, Guido Proietti



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Davide Bilò
Feliciano Colella
Luciano Gualà
Stefano Leucci
Guido Proietti

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Davide Bilò, Feliciano Colella, Luciano Gualà, Stefano Leucci, and Guido Proietti. An Improved Algorithm for Computing All the Best Swap Edges of a Tree Spanner. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 14:1-14:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ISAAC.2017.14

Abstract

A tree sigma-spanner of a positively real-weighted n-vertex and m-edge undirected graph G is a spanning tree T of G which approximately preserves (i.e., up to a multiplicative stretch factor sigma) distances in G. Tree spanners with provably good stretch factors find applications in communication networks, distributed systems, and network design. However, finding an optimal or even a good tree spanner is a very hard computational task. Thus, if one has to face a transient edge failure in T, the overall effort that has to be afforded to rebuild a new tree spanner (i.e., computational costs, set-up of new links, updating of the routing tables, etc.) can be rather prohibitive. To circumvent this drawback, an effective alternative is that of associating with each tree edge a best possible (in terms of resulting stretch) swap edge -- a well-established approach in the literature for several other tree topologies. Correspondingly, the problem of computing all the best swap edges of a tree spanner is a challenging algorithmic problem, since solving it efficiently means to exploit the structure of shortest paths not only in G, but also in all the scenarios in which an edge of T has failed. For this problem we provide a very efficient solution, running in O(n^2 log^4 n) time, which drastically improves (almost by a quadratic factor in n in dense graphs!) on the previous known best result.
Keywords
  • Transient edge failure
  • Swap algorithm
  • Tree spanner

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References

  1. Ittai Abraham, Yair Bartal, and Ofer Neiman. Embedding metrics into ultrametrics and graphs into spanning trees with constant average distortion. In Proc. of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 502-511, 2007. Google Scholar
  2. Stephen Alstrup, Jacob Holm, Kristian de Lichtenberg, and Mikkel Thorup. Maintaining information in fully dynamic trees with top trees. ACM Transactions on Algorithms, 1(2):243-264, 2005. URL: http://dx.doi.org/10.1145/1103963.1103966.
  3. Davide Bilò, Feliciano Colella, Luciano Gualà, Stefano Leucci, and Guido Proietti. A faster computation of all the best swap edges of a tree spanner. In Proc. of the 22nd Intl. Colloquium Structural Information and Communication Complexity, pages 239-253, 2015. URL: http://dx.doi.org/10.1007/978-3-319-25258-2_17.
  4. Davide Bilò, Feliciano Colella, Luciano Gualà, Stefano Leucci, and Guido Proietti. Effective edge-fault-tolerant single-source spanners via best (or good) swap edges. In Proc. of the 24th Intl. Colloquium Structural Information and Communication Complexity, in press. Google Scholar
  5. Davide Bilò, Luciano Gualà, and Guido Proietti. Finding best swap edges minimizing the routing cost of a spanning tree. Algorithmica, 68(2):337-357, 2014. URL: http://dx.doi.org/10.1007/s00453-012-9674-y.
  6. Davide Bilò, Luciano Gualà, and Guido Proietti. A faster computation of all the best swap edges of a shortest paths tree. Algorithmica, 73(3):547-570, 2015. URL: http://dx.doi.org/10.1007/s00453-014-9912-6.
  7. Andreas Brandstädt, Victor Chepoi, and Feodor F. Dragan. Distance approximating trees for chordal and dually chordal graphs. J. Algorithms, 30(1):166-184, 1999. URL: http://dx.doi.org/10.1006/jagm.1998.0962.
  8. Leizhen Cai and Derek G. Corneil. Tree spanners. SIAM J. Discrete Math., 8(3):359-387, 1995. URL: http://dx.doi.org/10.1137/S0895480192237403.
  9. Shantanu Das, Beat Gfeller, and Peter Widmayer. Computing all best swaps for minimum-stretch tree spanners. J. Graph Algorithms Appl., 14(2):287-306, 2010. Google Scholar
  10. Yuval Emek and David Peleg. Approximating minimum max-stretch spanning trees on unweighted graphs. SIAM J. Comput., 38(5):1761-1781, 2008. URL: http://dx.doi.org/10.1137/060666202.
  11. Harold N. Gabow. A scaling algorithm for weighted matching on general graphs. In Proc. of the 26th Annual Symposium on Foundations of Computer Science, pages 90-100, 1985. URL: http://dx.doi.org/10.1109/SFCS.1985.3.
  12. Giuseppe F. Italiano and Rajiv Ramaswami. Maintaining spanning trees of small diameter. Algorithmica, 22(3):275-304, 1998. URL: http://dx.doi.org/10.1007/PL00009225.
  13. Hiro Ito, Kazuo Iwama, Yasuo Okabe, and Takuya Yoshihiro. Polynomial-time computable backup tables for shortest-path routing. In Proc. of the 10th Intl. Colloquium Structural Information and Communication Complexity, pages 163-177, 2003. Google Scholar
  14. Camille Jordan. Sur les assemblages de lignes. J. Reine Angew. Math, 70(185):81, 1869. Google Scholar
  15. Christian Liebchen and Gregor Wünsch. The zoo of tree spanner problems. Discrete Applied Mathematics, 156(5):569-587, 2008. URL: http://dx.doi.org/10.1016/j.dam.2007.07.001.
  16. Enrico Nardelli, Guido Proietti, and Peter Widmayer. A faster computation of the most vital edge of a shortest path. Inf. Process. Lett., 79(2):81-85, 2001. URL: http://dx.doi.org/10.1016/S0020-0190(00)00175-7.
  17. Enrico Nardelli, Guido Proietti, and Peter Widmayer. Nearly linear time minimum spanning tree maintenance for transient node failures. Algorithmica, 40(2):119-132, 2004. URL: http://dx.doi.org/10.1007/s00453-004-1099-9.
  18. Seth Pettie. Sensitivity analysis of minimum spanning trees in sub-inverse-Ackermann time. In Proc. of the 16th Intl. Symposium on Algorithms and Computation, pages 964-973, 2005. URL: http://dx.doi.org/10.1007/11602613_96.
  19. Guido Proietti. Dynamic maintenance versus swapping: An experimental study on shortest paths trees. In Proc. of the 4th Intl. Workshop on Algorithm Engineering, pages 207-217, 2000. URL: http://dx.doi.org/10.1007/3-540-44691-5_18.
  20. Aleksej Di Salvo and Guido Proietti. Swapping a failing edge of a shortest paths tree by minimizing the average stretch factor. Theor. Comput. Sci., 383(1):23-33, 2007. URL: http://dx.doi.org/10.1016/j.tcs.2007.03.046.
  21. Robert Endre Tarjan. Sensitivity analysis of minimum spanning trees and shortest path trees. Inf. Process. Lett., 14(1):30-33, 1982. URL: http://dx.doi.org/10.1016/0020-0190(82)90137-5.
  22. Bang Ye Wu, Chih-Yuan Hsiao, and Kun-Mao Chao. The swap edges of a multiple-sources routing tree. Algorithmica, 50(3):299-311, 2008. URL: http://dx.doi.org/10.1007/s00453-007-9080-z.
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