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Smoothed Analysis

  • Reference work entry
  • First Online:
Encyclopedia of Algorithms

  • 187 Accesses

Years and Authors of Summarized Original Work

  • 2001; Spielman, Teng

  • 2004; Beier, Vöcking

Problem Definition

Smoothed analysis has originally been introduced by Spielman and Teng [22] in 2001 to explain why the simplex method is usually fast in practice despite its exponential worst-case running time. Since then it has been applied to a wide range of algorithms and optimization problem. In smoothed analysis, inputs are generated in two steps: first, an adversary chooses an arbitrary instance, and then this instance is slightly perturbed at random. The smoothed performance of an algorithm is defined to be the worst expected performance the adversary can achieve. This model can be viewed as a less pessimistic worst-case analysis, in which the randomness rules out pathological worst-case instances that are rarely observed in practice but dominate the worst-case analysis. If the smoothed running time of an algorithm is low (i.e., the algorithm is efficient in expectation on any perturbed...

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  1. Arthur D, Vassilvitskii S (2009) Worst-case and smoothed analysis of the ICP algorithm, with an application to the k-means method. SIAM J Comput 39(2):766–782

    Article  MathSciNet  MATH  Google Scholar 

  2. Arthur D, Manthey B, Röglin H (2011) Smoothed analysis of the k-means method. J ACM 58(5)

    Google Scholar 

  3. Banderier C, Beier R, Mehlhorn K (2003) Smoothed analysis of three combinatorial problems. In: Proceedings of the 28th international symposium on mathematical foundations of computer science (MFCS), Bratislava. Lecture notes in computer science, vol 2747. Springer, pp 198–207

    Google Scholar 

  4. Becchetti L, Leonardi S, Marchetti-Spaccamela A, Schäfer G, Vredeveld T (2006) Average case and smoothed competitive analysis of the multilevel feedback algorithm. Math Oper Res 31(1):85–108

    Article  MathSciNet  MATH  Google Scholar 

  5. Beier R, Vöcking B (2004) Random knapsack in expected polynomial time. J Comput Syst Sci 69(3):306–329

    Article  MathSciNet  MATH  Google Scholar 

  6. Beier R, Vöcking B (2006) Typical properties of winners and losers in discrete optimization. SIAM J Comput 35(4):855–881

    Article  MathSciNet  MATH  Google Scholar 

  7. Bläser M, Manthey B, Rao BVR (2011) Smoothed analysis of partitioning algorithms for Euclidean functionals. In: Proceedings of the 12th workshop on algorithms and data structures (WADS), New York. Lecture notes in computer science. Springer, pp 110–121

    Chapter  Google Scholar 

  8. Blum AL, Dunagan JD (2002) Smoothed analysis of the perceptron algorithm for linear programming. In: Proceedings of the 13th annual ACM-SIAM symposium on discrete algorithms (SODA), San Francisco. SIAM, pp 905–914

    Google Scholar 

  9. Boros E, Elbassioni K, Fouz M, Gurvich V, Makino K, Manthey B (2011) Stochastic mean payoff games: smoothed analysis and approximation schemes. In: Proceedings of the 38th international colloquium on automata, languages and programming (ICALP), Zurich, Part I. Lecture notes in computer science, vol 6755. Springer, pp 147–158

    Google Scholar 

  10. Brunsch T, Röglin H (2012) Improved smoothed analysis of multiobjective optimization. In: Proceedings of the 44th annual ACM symposium on theory of computing (STOC), New York, pp 407–426

    Google Scholar 

  11. Chen X, Deng X, Teng SH (2009) Settling the complexity of computing two-player Nash equilibria. J ACM 56(3)

    Google Scholar 

  12. Damerow V, Manthey B, auf der Heide FM, Räcke H, Scheideler C, Sohler C, Tantau T (2012) Smoothed analysis of left-to-right maxima with applications. ACM Trans Algorithms 8(3): Article no 30

    Google Scholar 

  13. Englert M, Röglin H, Vöcking B (2007) Worst case and probabilistic analysis of the 2-Opt algorithm for the TSP. In: Proceedings of the 18th annual ACM-SIAM symposium on discrete algorithms (SODA), New Orleans. SIAM, pp 1295–1304

    Google Scholar 

  14. Etscheid M, Röglin H (2014) Smoothed analysis of local search for the maximum-cut problem. In: Proceedings of the 25th annual ACM-SIAM symposium on discrete algorithms (SODA), Portland pp 882–889

    Google Scholar 

  15. Fouz M, Kufleitner M, Manthey B, Zeini Jahromi N (2012) On smoothed analysis of quicksort and Hoare’s find. Algorithmica 62(3–4):879–905

    Article  MathSciNet  MATH  Google Scholar 

  16. Manthey B, Reischuk R (2007) Smoothed analysis of binary search trees. Theor Comput Sci 378(3):292–315

    Article  MathSciNet  MATH  Google Scholar 

  17. Moitra A, O’Donnell R (2012) Pareto optimal solutions for smoothed analysts. SIAM J Comput 41(5):1266–1284

    Article  MathSciNet  MATH  Google Scholar 

  18. Röglin H, Teng SH (2009) Smoothed analysis of multiobjective optimization. In: Proceedings of the 50th annual IEEE symposium on foundations of computer science (FOCS), Atlanta. IEEE, pp 681–690

    Google Scholar 

  19. Sankar A, Spielman DA, Teng SH (2006) Smoothed analysis of the condition numbers and growth factors of matrices. SIAM J Matrix Anal Appl 28(2):446–476

    Article  MathSciNet  MATH  Google Scholar 

  20. Schäfer G, Sivadasan N (2005) Topology matters: smoothed competitiveness of metrical task systems. Theor Comput Sci 241(1–3):216–246

    Article  MathSciNet  MATH  Google Scholar 

  21. Spielman DA, Teng SH (2003) Smoothed analysis of termination of linear programming algorithms. Math Program 97(1–2):375–404

    MathSciNet  MATH  Google Scholar 

  22. Spielman DA, Teng SH (2004) Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. J ACM 51(3):385–463

    Article  MathSciNet  MATH  Google Scholar 

  23. Vershynin R (2009) Beyond Hirsch conjecture: walks on random polytopes and smoothed complexity of the simplex method. SIAM J Comput 39(2):646–678

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of Bonn, Bonn, Germany

    Heiko Röglin

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  1. Heiko Röglin
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Correspondence to Heiko Röglin .

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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Röglin, H. (2016). Smoothed Analysis. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_582

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