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Smoothed analysis: an attempt to explain the behavior of algorithms in practice

Authors:

Daniel A. Spielman,

Shang-Hua TengAuthors Info & Claims

Communications of the ACM, Volume 52, Issue 10

Pages 76 - 84

https://doi.org/10.1145/1562764.1562785

Published: 01 October 2009 Publication History

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Abstract

This Gödel Prize-winning work traces the steps toward modeling real data.

References

[1]

Adler, I. The expected number of pivots needed to solve parametric linear programs and the efficiency of the self-dual simplex method. Technical report, University of California at Berkeley (May 1983).

Google Scholar

[2]

Andoni, A. and Krauthgamer, R. The smoothed complexity of edit distance. In Proceedings of ICALP, volume 5125 of Lecture Notes in Computer Science (Springer, 2008) 357--369.

Digital Library

Google Scholar

[3]

Arthur, D. and Vassilvitskii, S. How slow is the k-means method? In SOCG '06, the 22nd Annual ACM Symposium on Computational Geometry (2006) 144--153.

Digital Library

Google Scholar

[4]

Arthur, D. and Vassilvitskii, S. Worst-case and smoothed analysis of the ICP algorithm, with an application to the k-means method. In FOCS '06, the 47th Annual IEEE Symposium on Foundations of Computer Science (2006) 153--164.

Digital Library

Google Scholar

[5]

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

Digital Library

Google Scholar

[6]

Beier, R. and Vöcking, B. Typical properties of winners and losers in discrete optimization. In STOC '04: the 36th Annual ACM Symposium on Theory of Computing (2004), 343--352.

Digital Library

Google Scholar

[7]

Blum, A. and Dunagan, J. Smoothed analysis of the perceptron algorithm for linear programming. In SODA '02 (2002), 905--914.

Digital Library

Google Scholar

[8]

Blum, A. and Spencer, J. Coloring random and semi-random k-colorable graphs. J. Algorithms 19, 2 (1995), 204--234.

Digital Library

Google Scholar

[9]

Bohman, T., Frieze, A. and Martin, R. How many random edges make a dense graph hamiltonian? Random Struct. Algorithms 22, 1 (2003), 33--42.

Digital Library

Google Scholar

[10]

Bürgisser, P., Cucker, F., and Lotz, M. Smoothed analysis of complex conic condition numbers. J. de Mathématiques Pures Appliqués 86 4 (2006), 293--309.

Google Scholar

[11]

Damerow, V., Meyer auf der Heide, F., Räcke, H., Scheideler, C., and Sohler, C. Smoothed motion complexity. In Proceedings of the 11th Annual European Symposium on Algorithms (2003), 161--171.

Google Scholar

[12]

Dantzig, G.B. Maximization of linear function of variables subject to linear inequalities. Activity Analysis of Production and Allocation. T.C. Koopmans, Ed. 1951, 339--347.

Google Scholar

[13]

Dunagan, J., Spielman, D.A., and Teng, S.-H. Smoothed analysis of condition numbers and complexity implications for linear programming. Mathematical Programming, Series A, 2009. To appear. Preliminary version available at http://arxiv.org/abs/cs/0302011v2.

Digital Library

Google Scholar

[14]

Edelman, A. Eigenvalue roulette and random test matrices. Linear Algebra for Large Scale and Real-Time Applications. Marc S. Moonen, Gene H. Golub, and Bart L. R. De Moor, eds. NATO ASI Series, 1992, 365--368.

Google Scholar

[15]

Englert, M., Röglin, H., and Vöcking, B. Worst case and probabilistic analysis of the 2-opt algorithm for the TSP: extended abstract. In SODA '07: The 18th Annual ACM-SIAM Symposium on Discrete Algorithms (2007), 1295--1304.

Digital Library

Google Scholar

[16]

Feige, U. Refuting smoothed 3CNF formulas. In The 48th Annual IEEE Symposium on Foundations of Computer Science (2007), 407--417.

Digital Library

Google Scholar

[17]

Flaxman, A. and Frieze, A.M. The diameter of randomly perturbed digraphs and some applications. In APPROX-RANDOM (2004), 345--356.

Digital Library

Google Scholar

[18]

Gass, S. and Saaty, T. The computational algorithm for the parametric objective function. Naval Res. Logist. Quart. 2, (1955), 39--45.

Crossref

Google Scholar

[19]

Haimovich, M. The simplex algorithm is very good!: On the expected number of pivot steps and related properties of random linear programs. Technical report, Columbia University (April 1983).

Google Scholar

[20]

Huang, L.-S. and Teng, S.-H. On the approximation and smoothed complexity of Leontief market equilibria. In Frontiers of Algorithms Workshop (2007), 96--107.

Digital Library

Google Scholar

[21]

Kalai, A. and Teng, S.-H. Decision trees are PAC - learnable from most product distributions: a smoothed analysis. ArXiv e-prints (December 2008).

Google Scholar

[22]

Karger, D. and Onak, K. Polynomial approximation schemes for smoothed and random instances of multidimensional packing problems. In SODA '07: the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (2007), 1207--1216.

Digital Library

Google Scholar

[23]

Kelner, J.A. and Nikolova, E. On the hardness and smoothed complexity of quasi-concave minimization. In The 48th Annual IEEE Symposium on Foundations of Computer Science (2007), 472--482.

Digital Library

Google Scholar

[24]

Kelner, J.A. and Spielman, D.A. A randomized polynomial-time simplex algorithm for linear programming. In The 38th Annual ACM Symposium on Theory of Computing (2006), 51--60.

Digital Library

Google Scholar

[25]

Klee, V. and Minty, G.J. How good is the simplex algorithm? Inequalities -- III. O. Shisha, ed. Academic Press, 1972, 159--175.

Google Scholar

[26]

Krivelevich, M., Sudakov, B. and Tetali, P. On smoothed analysis in dense graphs and formulas. Random Struct. Algorithms 29 (2005), 180--193.

Digital Library

Google Scholar

[27]

Ma, B. Why greed works for shortest common superstring problem. In CPM '08: Proceedings of the 19th Annual Symposium on Combinatorial Pattern Matching (2008), 244--254.

Digital Library

Google Scholar

[28]

Manthey, B. and Röglin, H. Improved smoothed analysis of the k-means method. In SODA '09 (2009).

Digital Library

Google Scholar

[29]

Mehlhorn, K. and Näher, S. The LEDA Platform of Combinatorial and Geometric Computing. Cambridge University Press, New York, 1999.

Digital Library

Google Scholar

[30]

Mitzenmacher, M. and Vadhan, S. Why simple hash functions work: exploiting the entropy in a data stream. In SODA '08: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms (2008), 746--755.

Digital Library

Google Scholar

[31]

Röglin, H. and Vöcking, B. Smoothed analysis of integer programming. Proceedings of the 11th International Conference on Integer Programming and Combinatorial Optimization. M. Junger and V. Kaibel, eds. Volume 3509 of Lecture Notes in Computer Science, Springer, 2005, 276--290.

Digital Library

Google Scholar

[32]

Rudelson, M. and Vershynin, R. The littlewood-offord problem and invertibility of random matrices. Adv. Math. 218, (June 2008), 600--633.

Crossref

Google Scholar

[33]

Sankar, A. Smoothed analysis of Gaussian elimination. Ph.D. Thesis, MIT, 2004.

Google Scholar

[34]

Sankar, A., Spielman, D.A., and Teng, S.-H. Smoothed analysis of the condition numbers and growth factors of matrices. SIAM J. Matrix Anal. Appl. 28, 2 (2006), 446--476.

Digital Library

Google Scholar

[35]

Spielman, D.A. and Teng, S.-H. Smoothed analysis of algorithms. In Proceedings of the International Congress of Mathematicians (2002), 597--606.

Google Scholar

[36]

Spielman, D.A. and Teng, S.-H. Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. J. ACM 51, 3 (20040, 385--463.

Digital Library

Google Scholar

[37]

Teng, S.-H. Algorithm design and analysis with perburbations. In Fourth International Congress of Chinese Mathematicans (2007).

Google Scholar

[38]

Vershynin, R. Beyond Hirsch conjecture: Walks on random polytopes and smoothed complexity of the simplex method. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (2006), 133--142.

Digital Library

Google Scholar

[39]

Vu, V.H. and Tao, T. The condition number of a randomly perturbed matrix. In STOC '07: the 39th Annual ACM Symposium on Theory of Computing (2007), 248--255.

Digital Library

Google Scholar

[40]

Wschebor, M. Smoothed analysis of κ(a). J. Complexity 20, 1 (February 2004), 97--107.

Digital Library

Google Scholar

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Index Terms

Smoothed analysis: an attempt to explain the behavior of algorithms in practice
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
    2. Graph theory
2. Theory of computation
  1. Design and analysis of algorithms

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Published In

Communications of the ACM Volume 52, Issue 10

A View of Parallel Computing

October 2009

134 pages

ISSN:0001-0782

EISSN:1557-7317

DOI:10.1145/1562764

Issue’s Table of Contents

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Published: 01 October 2009

Published in CACM Volume 52, Issue 10

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