research-article

Free access

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

All formats PDF

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

Cited By

View all

Díaz AMorris PPerarnau GSerra O(2024)Speeding up random walk mixing by starting from a uniform vertexElectronic Journal of Probability10.1214/24-EJP109129:noneOnline publication date: 1-Jan-2024
https://doi.org/10.1214/24-EJP1091
Giannakopoulos YKleinberg RSaban DBergemann D(2024)A Smoothed FPTAS for Equilibria in Congestion GamesProceedings of the 25th ACM Conference on Economics and Computation10.1145/3670865.3673615(401-413)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3670865.3673615
Dinitz MFineman JGilbert SNewport C(2024)Smoothed Analysis of Information Spreading in Dynamic NetworksJournal of the ACM10.1145/366183171:3(1-24)Online publication date: 11-Jun-2024
https://dl.acm.org/doi/10.1145/3661831
Show More Cited By

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

Recommendations

Steepest descent using smoothed gradients

This paper reviews solving differential equations using the least-squares steepest descent method with Sobolev gradients. The method's superiority over standard steepest descent with a Euclidean gradient is explained in terms of stability and the ...
Artificial intelligence researchers: Alan Turing, Marvin Minsky, Seymour Papert, Joseph Weizenbaum, Kevin Warwick, Ray Kurzweil
Smoothed particle hydrodynamics with diffusive flux for advection–diffusion equation with discontinuities
Abstract
The advection-diffusion equation (ADE) with variable diffusion coefficient can be written in a flux form to avoid rewriting the diffusion term with a drift. However, for solving the flux-form ADE using smoothed particle hydrodynamics (SPH), a ...
Highlights
- Reasons for non-physical oscillations in SPH with diffusive flux were analyzed.
- A new SPH with partial diffusive flux was proposed to eliminate the oscillations.
- Numerical error was reduced by at least one order of magnitude.
- ...

Comments

Information & Contributors

Information

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

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 October 2009

Published in CACM Volume 52, Issue 10

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Research-article
Popular
Refereed

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

114
Total Citations
View Citations
3,673
Total Downloads

Downloads (Last 12 months)458
Downloads (Last 6 weeks)84

Reflects downloads up to 28 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

View all

Díaz AMorris PPerarnau GSerra O(2024)Speeding up random walk mixing by starting from a uniform vertexElectronic Journal of Probability10.1214/24-EJP109129:noneOnline publication date: 1-Jan-2024
https://doi.org/10.1214/24-EJP1091
Giannakopoulos YKleinberg RSaban DBergemann D(2024)A Smoothed FPTAS for Equilibria in Congestion GamesProceedings of the 25th ACM Conference on Economics and Computation10.1145/3670865.3673615(401-413)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3670865.3673615
Dinitz MFineman JGilbert SNewport C(2024)Smoothed Analysis of Information Spreading in Dynamic NetworksJournal of the ACM10.1145/366183171:3(1-24)Online publication date: 11-Jun-2024
https://dl.acm.org/doi/10.1145/3661831
Peca-Medlin J(2024)Growth Factors of Orthogonal Matrices and Local Behavior of Gaussian Elimination with Partial and Complete PivotingSIAM Journal on Matrix Analysis and Applications10.1137/23M159773345:3(1599-1620)Online publication date: 22-Aug-2024
https://doi.org/10.1137/23M1597733
Gavorová ZSeidel MTouati Y(2024)Topological obstructions to quantum computation with unitary oraclesPhysical Review A10.1103/PhysRevA.109.032625109:3Online publication date: 28-Mar-2024
https://doi.org/10.1103/PhysRevA.109.032625
Abbas AAmbainis AAugustino BBärtschi ABuhrman HCoffrin CCortiana GDunjko VEgger DElmegreen BFranco NFratini FFuller BGacon JGonciulea CGribling SGupta SHadfield SHeese RKircher GKleinert TKoch TKorpas GLenk SMarecek JMarkov VMazzola GMensa SMohseni NNannicini GO’Meara CTapia EPokutta SProissl MRebentrost PSahin ESymons BTornow SValls VWoerner SWolf-Bauwens MYard JYarkoni SZechiel DZhuk SZoufal C(2024)Challenges and opportunities in quantum optimizationNature Reviews Physics10.1038/s42254-024-00770-96:12(718-735)Online publication date: 28-Oct-2024
https://doi.org/10.1038/s42254-024-00770-9
Escudero LFeldheim NKoliander GRomero J(2024)Efficient Computation of the Zeros of the Bargmann Transform Under Additive White NoiseFoundations of Computational Mathematics10.1007/s10208-022-09593-524:1(279-312)Online publication date: 1-Feb-2024
https://dl.acm.org/doi/10.1007/s10208-022-09593-5
Goldwasser SGruber DKalai AParadise OOh ANaumann TGloberson ASaenko KHardt MLevine S(2023)A theory of unsupervised translation motivated by understanding animal communicationProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3667743(37286-37320)Online publication date: 10-Dec-2023
https://dl.acm.org/doi/10.5555/3666122.3667743
Cade CFolkertsma MNiesen IWeggemans J(2023)Quantifying Grover speed-ups beyond asymptotic analysisQuantum10.22331/q-2023-10-10-11337(1133)Online publication date: 10-Oct-2023
https://doi.org/10.22331/q-2023-10-10-1133
Xia LWilliams BChen YNeville J(2023)Semi-random impossibilities of condorcet criterionProceedings of the Thirty-Seventh AAAI Conference on Artificial Intelligence and Thirty-Fifth Conference on Innovative Applications of Artificial Intelligence and Thirteenth Symposium on Educational Advances in Artificial Intelligence10.1609/aaai.v37i5.25727(5867-5875)Online publication date: 7-Feb-2023
https://dl.acm.org/doi/10.1609/aaai.v37i5.25727
Show More Cited By

View Options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Digital Edition

View this article in digital edition.

Digital Edition

Magazine Site

View this article on the magazine site (external)

Magazine Site

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Abstract

References

Cited By

Index Terms

Recommendations

Steepest descent using smoothed gradients

Artificial intelligence researchers: Alan Turing, Marvin Minsky, Seymour Papert, Joseph Weizenbaum, Kevin Warwick, Ray Kurzweil

Smoothed particle hydrodynamics with diffusive flux for advection–diffusion equation with discontinuities

Comments

Information

Published In

Publisher

Publication History

Permissions

Check for updates

Qualifiers

Contributors

Other Metrics

Bibliometrics

Article Metrics

Other Metrics

Citations

Cited By

View options

PDF

eReader

Digital Edition

Magazine Site

Login options

Full Access

Share

Share this Publication link

Share on social media

Affiliations