Smoothed Analysis of Three Combinatorial Problems

  • Cyril Banderier
  • René Beier
  • Kurt Mehlhorn
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2747)

Abstract

Smoothed analysis combines elements over worst-case and average case analysis. For an instance x, the smoothed complexity is the average complexity of an instance obtained from x by a perturbation. The smoothed complexity of a problem is the worst smoothed complexity of any instance. Spielman and Teng introduced this notion for continuous problems. We apply the concept to combinatorial problems and study the smoothed complexity of three classical discrete problems: quicksort, left-to-right maxima counting, and shortest paths.

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References

  1. 1.
    Ahuja, R.K., Mehlhorn, K., Orlin, J.B., Tarjan, R.E.: Faster algorithms for the shortest path problem. J. Assoc. Comput. Mach. 37(2), 213–223 (1990)MATHMathSciNetGoogle Scholar
  2. 2.
    Becchetti, L., Leonardi, S., Marchetti-Spaccamela, A., Schäfer, G., Vredeveld, T.: Smoothening helps: A probabilistic analysis of the Multi-Level Feedback algorithm. (2003) (submitted) Google Scholar
  3. 3.
    Blum, A., Dunagan, J.D.: Smoothed analysis of the perceptron algorithm for linear programming. In: Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2002), pp. 905–914. ACM Press, New York (2002)Google Scholar
  4. 4.
    Cherkassky, B.V., Goldberg, A.V., Silverstein, C.: Buckets, heaps, lists, and monotone priority queues. SIAM J. Comput. 28(4), 1326–1346 (1999) (electronic)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dunagan, J.D., Spielman, D.A., Teng, S.-H.: Smoothed analysis of the renegar’s condition number for linear programming. In: SIAM Conference on Optimization (2002)Google Scholar
  6. 6.
    Goldberg, A.V.: A simple shortest path algorithm with linear average time. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 230–241. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    Gurevich, Y.: Average case completeness. Journal of Computer and System Sciences 42(3), 346–398 (1991)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hagerup, T.: Improved shortest paths on the word RAM. In: Automata, languages and programming (ICALP 2000), pp. 61–72. Springer, Berlin (2000)CrossRefGoogle Scholar
  9. 9.
    Levin, L.A.: Average case complete problems. SIAM J. Comput. 15(1), 285–286 (1986)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Meyer, U.: Shortest-Paths on arbitrary directed graphs in linear Average-Case time. In: Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), pp. 797–806. ACM Press, New York (2001)Google Scholar
  11. 11.
    Motwani, R., Raghavan, P.: Randomized algorithms. Cambridge University Press, Cambridge (1995)MATHGoogle Scholar
  12. 12.
    Raman, R.: Recent results on the single-source shortest paths problem. In: Blikle, A. (ed.) MFCS 1974. LNCS, vol. 28, pp. 61–72. Springer, Heidelberg (1997)Google Scholar
  13. 13.
    Spielman, D.A., Teng, S.-H.: Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. In: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, pp. 296–3051 (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Cyril Banderier
    • 1
  • René Beier
    • 2
  • Kurt Mehlhorn
    • 2
  1. 1.Laboratoire d’Informatique de Paris Nord Institut GaliléeUniversité Paris 13VilletaneuseFrance
  2. 2.Max-Planck-Institut für InformatikSaarbrückenGermany

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