Skip to main content
SpringerLink
Log in

An Overview of Randomized Algorithms

  • Chapter
Probabilistic Methods for Algorithmic Discrete Mathematics

Part of the book series: Algorithms and Combinatorics ((AC,volume 16))

  • 2103 Accesses

Abstract

A randomized algorithm makes random choices during its execution. The behavior of such an algorithm may thus be random even on a fixed input. The process of designing and analyzing a randomized algorithm focuses on establishing that it is likely to behave “well” on every input. The likelihood in such a statement depends only on the probabilistic choices made by the algorithm during execution and not on any assumptions about the input. It is especially important to distinguish a randomized algorithm from the averagecase analysis of algorithms, where one analyzes an algorithm assuming that its input is drawn from a fixed probability distribution. With a randomized algorithm, in contrast, no assumption is made about the input.

Supported by an Alfred P. Sloan Research Fellowship, an IBM Faculty Partnership Award, an ARO MURI Grant DAAH04-96-1-0007, and NSF Young Investigator Award CCR-9357849, with matching funds from IBM, Schlumberger Foundation, Shell Foundation, and Xerox Corporation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aleliunas R, Karp R.M., Lipton R.J., Lovâsz L., and Rackoff C, (1979): Random walks, universal traversal sequences, and the complexity of maze problems, in Proceedings of the 20th Annual Symposium on Foundations of Computer Science, pages 218–223, San Juan, Puerto Rico, October.

    Google Scholar 

  2. Aragon C.R. and Seidel R.G. (1989): Randomized search trees, in Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science, pages 540–545.

    Google Scholar 

  3. Ben-David S., Borodin A., Karp R.M., Tardos G., and Wigderson A. (1994): On the power of randomization in on-line algorithms, Algorithmica 11, (1), 2–14.

    Article  MathSciNet  MATH  Google Scholar 

  4. Blum M. and Kannan S (1989): Designing programs that check their work, in Proceedings of the 21st Annual ACM Symposium on Theory of Computing, pages 86–97. ACM.

    Google Scholar 

  5. Blum A. and Raghavan P. (1998): A theory of computing symposia, FUN with Algorithms.

    Google Scholar 

  6. Borodin A.B. and El Yaniv R. (1998): On-line algorithms, Cambridge University Press.

    Google Scholar 

  7. Coppersmith D. and Winograd S. (1990): Matrix multiplication via arithmetic progressions, Journal of Symbolic Computation 9, 251–280.

    Article  MathSciNet  MATH  Google Scholar 

  8. DeMillo R.A. and Lipton R.J. (1978): A probabilistic remark on algebraic program testing, Information Processing Letters 7, 193–195.

    Article  MATH  Google Scholar 

  9. Edmonds J. (1967): Systems of distinct representatives and linear algebra, Journal of Research of the National Bureau of Standards, 71B 4, 241–245.

    MathSciNet  Google Scholar 

  10. Feder T. and Motwani R. (1991): Clique partitions, graph compression and speeding-up algorithms, in Proceedings of the 25th Annual ACM Symposium on Theory of Computing, pages 123–133.

    Google Scholar 

  11. Floyd R.W. and Rivest R.L. (1975): Expected time bounds for selection, Communications of the ACM1 8, 165–172.

    Article  Google Scholar 

  12. Freivalds R. (1977): Probabilistic machines can use less running time, in B. Gilchrist, editor, Information Processing 77, Proceedings of IFIP Congress 77, pages 839–842, Amsterdam, August. North-Holland Publishing Company.

    Google Scholar 

  13. Goemans M.X. and Williamson D.P. (1994): 0.878-approximation algorithms for MAX-CUT and MAX-2SAT, in Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 422–431.

    Google Scholar 

  14. Hoare C.A.R. (1962): Quicksort, Computer Journal 5, 10–15.

    Article  MathSciNet  MATH  Google Scholar 

  15. Hoperoft J.E. and Karp R.M. (1973): An n5/2 algorithm for maximum matching in bipartite graphs, SIAM Journal on Computing 2, 225–231.

    Article  MathSciNet  Google Scholar 

  16. Karger D.R. (1993): Global min-cuts in RJVC, and other ramifications of a simple min-cut algorithm, in Proceedings of the 4th Annual ACM-SIAM Symposium on Discrete Algorithms.

    Google Scholar 

  17. Karger D.R., Klein P.N. and Tarjan R.E. (1995): A randomized linear-time algorithm for finding minimum spanning trees, Journal of the ACM 42, 321–328.

    Article  MathSciNet  MATH  Google Scholar 

  18. Karger D., Motwani R. and Sudan M. (1994): Approximate graph coloring by semidefinite programming, in Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science, 2–13.

    Google Scholar 

  19. Karp R.M. (1991): An introduction to randomized algorithms, Discrete Applied Mathematics 34, 165–201.

    Article  MathSciNet  MATH  Google Scholar 

  20. Karp R.M., Upfal E. and Wigderson A. (1986): Constructing a perfect matching is in random NC, Combinatorica 6, 35–48.

    Article  MathSciNet  MATH  Google Scholar 

  21. Karp R.M., Upfal E. and Wigderson A. (1988): The complexity of parallel search, Journal of Computer and System Sciences 36, 225–253.

    Article  MathSciNet  MATH  Google Scholar 

  22. Lovâsz L. (1979): On determinants, matchings and random algorithms, in L. Budach, editor, Fundamentals of Computing Theory. Akademia-Verlag, Berlin.

    Google Scholar 

  23. Maffioli F., Speranza M.G. and Vercellis C. (1985): Randomized algorithms, in M. O’hEigertaigh, J.K. Lenstra, and A.H.G. Rinooy Kan, editors, Combinatorial Optimization: Annotated Bibliographies, 89–105. John Wiley and Sons, New York.

    Google Scholar 

  24. Micali S. and Vazirani V.V. (1980): An 0(.1-iVilei) algorithm for finding maximum matching in general graphs, in Proceedings of the 21st Annual IEEE Symposium on Foundations of Computer Science, 17–27.

    Google Scholar 

  25. Motwani R., NaorJ. and Raghavan P. (1996): Randomization in approximation algorithms, in D. Hochbaum, editor, Approximation Algorithms, PWS.

    Google Scholar 

  26. Motwani R. and Raghavan P. (1995): Randomized Algorithms, Cambridge University Press, New York.

    MATH  Google Scholar 

  27. Mulmuley K. (1993): Computational Geometry: An Introduction Through Randomized Algorithms, Prentice Hall, New York.

    MATH  Google Scholar 

  28. Mulmuley K., Vazirani U.V. and Vazirani V.V. (1987): Matching is as easy as matrix inversion, Combinatorica 7, 105–113.

    Article  MathSciNet  MATH  Google Scholar 

  29. Pugh W. (1990): Skip lists: A probabilistic alternative to balanced trees, Communications of the ACM 33 (6), 668–676.

    Article  MathSciNet  Google Scholar 

  30. Rabin M.O. (1980): Probabilistic algorithm for testing primality, Journal of Number Theory 12, 128–138.

    Article  MathSciNet  MATH  Google Scholar 

  31. Rabin M.O. (1983): Randomized Byzantine generals, in Proceedings of the 24th Annual Symposium on Foundations of Computer Science, 403–409.

    Google Scholar 

  32. Rabin M.O. and Vazirani V.V. (1984): Maximum matchings in general graphs through randomization, Technical Report TR-15–84, Aiken Computation Laboratory, Harvard University.

    Google Scholar 

  33. Rabin M.O. and Vazirani V.V. (1989): Maximum matchings in general graphs through randomization, Journal of Algorithms 10, 557–567.

    Article  MathSciNet  MATH  Google Scholar 

  34. Saks M. and Wigderson A. (1986): Probabilistic Boolean decision trees and the complexity of evaluating game trees, in Proceedings of the 27th Annual IEEE Symposium on Foundations of Computer Science, 29–38, Toronto, Ontario.

    Google Scholar 

  35. Schrijver A. (1986): Theory of Linear and Integer Programming, John Wiley, New York.

    MATH  Google Scholar 

  36. Schwartz J.T. (1980): Fast probabilistic algorithms for verification of polynomial identities, Journal of the ACM 27(4), 701–717, October.

    Google Scholar 

  37. Seidel R.G. (1991): Small-dimensional linear programming and convex hulls made easy, Discrete and Computational Geometry 6, 423–434.

    Article  MathSciNet  MATH  Google Scholar 

  38. Sinclair A. (1992): Algorithms for Random Generation and Counting: A Markov Chain Approach, Progress in Theoretical Computer Science, Birkhauser, Boston.

    Google Scholar 

  39. Snir M. (1985): Lower bounds on probabilistic linear decision trees, Theoretical Computer Science 38, 69–82.

    Article  MathSciNet  MATH  Google Scholar 

  40. Solovay R. and Strassen V. (1977): A fast Monte-Carlo test for primality, SIAM Journal on Computing, 6(1), 84–85, March, see also SIAM Journal on Computing 7,1 February 1978, 118.

    Google Scholar 

  41. Tarsi M. (1983): Optimal search on some game trees, Journal of the ACM 30, 389–396.

    Article  MathSciNet  MATH  Google Scholar 

  42. Tutte W.T. (1947): The factorization of linear graphs, J. of the London Math. Soc. 22, 107–111.

    Article  MathSciNet  MATH  Google Scholar 

  43. Valiant L.G. (1982): A scheme for fast parallel communication, SIAM Journal on Computing 11, 350–361.

    Article  MathSciNet  MATH  Google Scholar 

  44. Vazirani V.V. (1994): A theory of alternating paths and blossoms for proving correctness of O(VVE) graph maximum matching algorithms, Combinatorica 14 (1), 71–109.

    Article  MathSciNet  MATH  Google Scholar 

  45. Welsh D.J.A. (1983): Randomised algorithms, Discrete Applied Mathematics 51, 33–145.

    MathSciNet  Google Scholar 

  46. Yao A.C.-C. (1977): Probabilistic computations: Towards a unified measure of complexity, in Proceedings of the 17th Annual Symposium on Foundations of Computer Science, pages 222–227.

    Google Scholar 

  47. Zippel R.E. (1979): Probabilistic algorithms for sparse polynomials, in Proceedings of EUROSAM 79 72 of Lecture Notes in Computer Science, pages 216–226, Marseille.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Stanford University, CA, 94305, USA

    Rajeev Motwani

  2. IBM Almaden Research Center, 650 Harry Road, San Jose, CA, 95120, USA

    Prabhakar Raghavan

Authors
  1. Rajeev Motwani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Prabhakar Raghavan
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

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

    Michel Habib

  2. Department of Statistics, University of Oxford, 1 South Parks Road, OX1 3TG, Oxford, UK

    Colin McDiarmid

  3. Equipe Combinatoire, Université Pierre et Marie Curie, Paris 6 Case 189, 4, place Jussieu, 75252, Paris Cedex 5, France

    Jorge Ramirez-Alfonsin  & Bruce Reed  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Motwani, R., Raghavan, P. (1998). An Overview of Randomized Algorithms. In: Habib, M., McDiarmid, C., Ramirez-Alfonsin, J., Reed, B. (eds) Probabilistic Methods for Algorithmic Discrete Mathematics. Algorithms and Combinatorics, vol 16. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-12788-9_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-12788-9_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08426-3

  • Online ISBN: 978-3-662-12788-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation