Advertisement

Finding Best Swap Edges Minimizing the Routing Cost of a Spanning Tree

  • Davide Bilò
  • Luciano Gualà
  • Guido Proietti
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6281)

Abstract

Given an n-node, undirected and 2-edge-connected graph G = (V,E) with positive real weights on its m edges, given a set of k source nodes S ⊆ V, and given a spanning tree T of G, the routing cost of T w.r.t. S is the sum of the distances in T from every source s ∈ S to all the other nodes of G. If an edge e of T undergoes a transient failure and connectivity needs to be promptly reestablished, then to reduce set-up and rerouting costs it makes sense to temporarily replace e by means of a swap edge, i.e., an edge in G reconnecting the two subtrees of T induced by the removal of e. Then, a best swap edge for e is a swap edge which minimizes the routing cost of the tree obtained after the swapping. As a natural extension, the all-best swap edges problem is that of finding a best swap edge for every edge of T. Such a problem has been recently solved in O(mn) time and linear space for arbitrary k, and in O(n 2 + mlogn) time and O(n 2) space for the special case k = 2. In this paper, we are interested to the prominent cases k = O(1) and k = n, which model realistic communication paradigms. For these cases, we present a linear space and \(\widetilde O(m)\) time algorithm, and thus we improve both the above running times (but for quite dense graphs in the case k = 2, for which however it is noticeable we make use of only linear space). Moreover, we provide an accurate analysis showing that when k = n, the obtained swap tree is effective in terms of routing cost.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ackermann, W.: Zum Hilbertschen Aufbau der reellen Zahlen. Mathematical Annals 99, 118–133 (1928)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Das, S., Gfeller, B., Widmayer, P.: Computing best swaps in optimal tree spanners. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 716–727. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Di Salvo, A., Proietti, G.: Swapping a failing edge of a shortest paths tree by minimizing the average stretch factor. Theoretical Computer Science 383(1), 23–33 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Flocchini, P., Mesa Enriques, A., Pagli, L., Prencipe, G., Santoro, N.: Point-of-failure shortest-path rerouting: computing the optimal swap edges distributively. IEICE Transactions 89-D(2), 700–708 (2006)CrossRefGoogle Scholar
  5. 5.
    Gfeller, B.: Faster swap edge computation in minimum diameter spanning trees. In: Halperin, D., Mehlhorn, K. (eds.) ESA 2008. LNCS, vol. 5193, pp. 454–465. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Gfeller, B., Santoro, N., Widmayer, P.: A distributed algorithm for finding all best swap edges of a minimum diameter spanning tree. In: Pelc, A. (ed.) DISC 2007. LNCS, vol. 4731, pp. 268–282. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Harel, D., Tarjan, R.E.: Fast algorithms for finding nearest common ancestors. SIAM Journal on Computing 13(2), 338–355 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hershberger, J.: Finding the upper envelope of n line segments in O(n logn) time. Information Processing Letters 33(4), 169–174 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Johnson, D.S., Lenstra, J.K., Rinnooy Kan, A.H.G.: The complexity of the network design problem. Networks 8, 279–285 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Nardelli, E., Proietti, G., Widmayer, P.: Finding all the best swaps of a minimum diameter spanning tree under transient edge failures. J. Graph Algorithms and Applications 5(5), 39–57 (2001)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Nardelli, E., Proietti, G., Widmayer, P.: Swapping a failing edge of a single source shortest paths tree is good and fast. Algorithmica 35(1), 56–74 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Nivasch, G.: Improved bounds and new techniques for Davenport–Schinzel sequences and their generalizations. J. ACM 57(3), 1–44 (2010)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Pettie, S.: Sensitivity analysis of minimum spanning trees in sub-inverse-Ackermann time. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 964–973. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    Wong, R.: Worst-case analysis of network design problem heuristics. SIAM J. Algebric Discrete Methods 1, 51–63 (1980)zbMATHCrossRefGoogle Scholar
  15. 15.
    Wu, B.Y., Hsiao, C.-Y., Chao, K.-M.: The swap edges of a multiple-sources routing tree. Algorithmica 50(3), 299–311 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Wu, B.Y., Lancia, G., Bafna, V., Chao, K.-M., Ravi, R., Tang, C.Y.: A polynomial-time approximation scheme for minimum routing cost spanning trees. SIAM J. Computing 29(3), 761–778 (1999)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Wu, B.Y., Chao, K.-M., Tang, C.Y.: Approximation algorithms for some optimum communication spanning tree problems. Discrete Applied Mathematics 102, 245–266 (2000)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Davide Bilò
    • 1
  • Luciano Gualà
    • 2
  • Guido Proietti
    • 1
    • 3
  1. 1.Dipartimento di InformaticaUniversità di L’AquilaL’AquilaItaly
  2. 2.Dipartimento di MatematicaUniversità di Tor VergataRomaItaly
  3. 3.Istituto di Analisi dei Sistemi ed Informatica, CNRRomaItaly

Personalised recommendations