Finding the Most Vital Node of a Shortest Path

  • Enrico Nardelli
  • Guido Proietti
  • Peter Widmayer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2108)


In an undirected, 2-node connected graph G = (V,E) with positive real edge lengths, the distance between any two nodes r and s is the length of a shortest path between r and s in G. The removal of a node and its incident edges from G mayincrease the distance from r to s. A most vital node of a given shortest path from r to s is a node (other than r and s) whose removal from G results in the largest increase of the distance from r to s. In the past, the problem of finding a most vital node of a given shortest path has been studied because of its implications in network management, where it is important to know in advance which component failure will affect network efficiency the most. In this paper, we show that this problem can be solved in O(m + n log n) time and O(m) space, where m and n denote the number of edges and the number of nodes in G.


Short Path Incident Edge Node Failure Short Path Tree Short Path Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    M.O. Ball, B.L. Golden and R.V. Vohra, Finding the most vital arcs in a network, Oper. Res. Letters, 8 (1989) 73–76.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A. Bar-Noy, S. Khuller and B. Schieber, The complexity of finding most vital arcs and nodes. TR CS-TR-3539, Institute for Advanced Studies, Universityof Maryland, College Park, MD, 1995.Google Scholar
  3. 3.
    H.W. Corleyand D.Y. Sha, Most vital links and nodes in weighted networks, Oper. Res. Letters, 1 (1982) 157–160.CrossRefGoogle Scholar
  4. 4.
    E.W. Dijkstra, A note on two problems in connection with graphs, Numer. Math., 1 (1959) 269–271.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    M.L. Fredman and R.E. Tarjan, Fibonacci heaps and their uses in improved network optimization algorithms, Journal of the ACM, 34(3) (1987) 596–615.CrossRefMathSciNetGoogle Scholar
  6. 6.
    K. Malik, A.K. Mittal and S.K. Gupta, The k most vital arcs in the shortest path problem, Oper. Res. Letters, 8 (1989) 223–227.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    E. Nardelli, G. Proietti and P. Widmayer, Finding the detour-critical edge of a shortest path between two nodes, Info. Proc. Letters,67(1) (1998) 51–54.CrossRefMathSciNetGoogle Scholar
  8. 8.
    E. Nardelli, G. Proietti and P. Widmayer, A faster computation of the most vital edge of a shortest path between two nodes, Info. Proc. Letters, to appear. Also available as TR 15-99, Dipartimento di Matematica Pura ed Applicata, University of L’Aquila, L’Aquila, Italy, April 1999.Google Scholar
  9. 9.
    N. Nisan, Algorithms for selfish agents, Proc. of the 16th Symp. on Theoretical Aspects of Computer Science (STACS’99), Lecture Notes in Computer Science, Vol. 1563, Springer, (1999) 1–15.Google Scholar
  10. 10.
    N. Nisan and A. Ronen, Algorithmic mechanism design, Proc. of the 31st Annual ACM Symposium on Theory of Computing (STOC’99), (1999) 129–140.Google Scholar
  11. 11.
    J.S. Rosenschein and G. Zlotkin, Rules of Encounter: Designing Conventions for Automated Negotiation Among Computers. MIT Press, Cambridge, Massachusetts, 1994.Google Scholar
  12. 12.
    R.E. Tarjan, Efficiencyof a good but not linear set union algorithm, Journal of the ACM, 22 (1975) 215–225.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    S. Venema, H. Shen and F. Suraweera, A parallel algorithm for the single most vital vertex problem with respect to single source shortest paths, Online Proc. of the First Int. Conf. on Parallel and Distributed Computing, Applications and Technologies (PDCAT’2000), Chapter 22,

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Enrico Nardelli
    • 1
    • 2
  • Guido Proietti
    • 1
    • 2
  • Peter Widmayer
    • 3
  1. 1.Dipartimento di Matematica Pura ed ApplicataUniversitá di L’AquilaL’AquilaItaly
  2. 2.Istituto di Analisi dei Sistemi ed InformaticaConsiglio Nazionale delle RicercheRomaItaly
  3. 3.Institut für Theoretische InformatikETH ZentrumZürichSwitzerland

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