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An Even Simpler Linear-Time Algorithm for Verifying Minimum Spanning Trees

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Graph-Theoretic Concepts in Computer Science (WG 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5911))

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Abstract

A new linear-time algorithm is presented for the tree-path-maxima problem of, given a tree T with real edge weights and a list of pairs of distinct nodes in T, computing for each pair (u,v) on the list a maximum-weight edge on the path in T between u and v. Linear-time algorithms for the tree-path-maxima problem were known previously, but the new algorithm may be considered significantly simpler than the earlier solutions. A linear-time algorithm for the tree-path-maxima problem implies a linear-time algorithm for the MST-verification problem of determining whether a given spanning tree of a given undirected graph G with real edge weights is a minimum-weight spanning tree of G.

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References

  1. Ahuja, R.K., Magnanti, T.L., Orlin, J.R.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Upper Saddle River (1993)

    Google Scholar 

  2. Borůvka, O.: O jistém problému minimálním. Práce Mor. Přírodověd. Spol. v Brně 3, 37–58 (1926)

    Google Scholar 

  3. Buchsbaum, A.L., Georgiadis, L., Kaplan, H., Rogers, A., Tarjan, R.E., Westbrook, J.R.: Linear-time algorithms for dominators and other path-evaluation problems. SIAM J. Comput. 38, 1533–1573 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chazelle, B.: Computing on a free tree via complexity-preserving mappings. Algorithmica 2, 337–361 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chazelle, B.: A minimum spanning tree algorithm with inverse-Ackermann type complexity. J. Assoc. Comput. Mach. 47, 1028–1047 (2000)

    MATH  MathSciNet  Google Scholar 

  6. Chazelle, B., Rosenberg, B.: The complexity of computing partial sums off-line. Internat. J. Comput. Geometry Appl. 1, 33–45 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chiang, Y., Goodrich, M.T., Grove, E.F., Tamassia, R., Vengroff, D.E., Vitter, J.S.: External-memory graph algorithms. In: Proc. 6th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 139–149 (1995)

    Google Scholar 

  8. Dixon, B., Rauch, M., Tarjan, R.E.: Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM J. Comput. 21, 1184–1192 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fredman, M.L., Willard, D.E.: Trans-dichotomous algorithms for minimum spanning trees and shortest paths. J. Comput. System Sci. 48, 533–551 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  10. Graham, R.L., Hell, P.: On the history of the minimum spanning tree problem. Annals Hist. Comput. 7, 43–57 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  11. Karger, D.R., Klein, P.N., Tarjan, R.E.: A randomized linear-time algorithm to find minimum spanning trees. J. Assoc. Comput. Mach. 42, 321–328 (1995)

    MATH  MathSciNet  Google Scholar 

  12. Katriel, I., Sanders, P., Träff, J.L.: A practical minimum spanning tree algorithm using the cycle property. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 679–690. Springer, Heidelberg (2003)

    Google Scholar 

  13. King, V.: A simpler minimum spanning tree verification algorithm. Algorithmica 18, 263–270 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  14. Komlós, J.: Linear verification for spanning trees. Combinatorica 5, 57–65 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  15. Mareš, M.: The saga of minimum spanning trees. Comput. Sci. Rev. 2, 165–221 (2008)

    Article  Google Scholar 

  16. Pettie, S.: An inverse-Ackermann type lower bound for online minimum spanning tree verification. Combinatorica 26, 207–230 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Pettie, S., Ramachandran, V.: An optimal minimum spanning tree algorithm. J. Assoc. Comput. Mach. 49, 16–34 (2002)

    MathSciNet  Google Scholar 

  18. Pettie, S., Ramachandran, V.: Randomized minimum spanning tree algorithms using exponentially fewer random bits. ACM Trans. Algorithms 4(1), 5:1–5:27 (2008)

    Article  MathSciNet  Google Scholar 

  19. Tarjan, R.E.: Applications of path compression on balanced trees. J. Assoc. Comput. Mach. 26, 690–715 (1979)

    MATH  MathSciNet  Google Scholar 

  20. Yao, A.C.: Space-time tradeoff for answering range queries. In: Proc. 14th Annual ACM Symposium on Theory of Computing (STOC), pp. 128–136 (1982)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institut für Informatik, Universität Augsburg, 86135, Augsburg, Germany

    Torben Hagerup

Authors
  1. Torben Hagerup
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Editor information

Editors and Affiliations

  1. CNRS - LIRMM, 161 rue Ada, 34392, Montpellier Cedex 5, France

    Christophe Paul

  2. LIAFA, case 7014, Université Paris Diderot, 75205, Paris Cedex 13, France

    Michel Habib

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Hagerup, T. (2010). An Even Simpler Linear-Time Algorithm for Verifying Minimum Spanning Trees. In: Paul, C., Habib, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2009. Lecture Notes in Computer Science, vol 5911. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11409-0_16

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  • DOI: https://doi.org/10.1007/978-3-642-11409-0_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-11408-3

  • Online ISBN: 978-3-642-11409-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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