Abstract
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.
This work has been partiallysupp orted bythe Research Training Network contract no. HPRN-CT-1999-00104 funded bythe European Union, and bythe Research Project REACTION, partiallyfunded bythe Italian Ministryof University.
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References
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.
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.
H.W. Corleyand D.Y. Sha, Most vital links and nodes in weighted networks, Oper. Res. Letters, 1 (1982) 157–160.
E.W. Dijkstra, A note on two problems in connection with graphs, Numer. Math., 1 (1959) 269–271.
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.
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.
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.
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.
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.
N. Nisan and A. Ronen, Algorithmic mechanism design, Proc. of the 31st Annual ACM Symposium on Theory of Computing (STOC’99), (1999) 129–140.
J.S. Rosenschein and G. Zlotkin, Rules of Encounter: Designing Conventions for Automated Negotiation Among Computers. MIT Press, Cambridge, Massachusetts, 1994.
R.E. Tarjan, Efficiencyof a good but not linear set union algorithm, Journal of the ACM, 22 (1975) 215–225.
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, http://www2.comp.polyu.edu.hk/PDCAT2000/publish.html.
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Nardelli, E., Proietti, G., Widmayer, P. (2001). Finding the Most Vital Node of a Shortest Path. In: Wang, J. (eds) Computing and Combinatorics. COCOON 2001. Lecture Notes in Computer Science, vol 2108. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44679-6_31
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DOI: https://doi.org/10.1007/3-540-44679-6_31
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