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Concentration

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Probabilistic Methods for Algorithmic Discrete Mathematics

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

Summary

Upper bounds on probabilities of large deviations for sums of bounded independent random variables may be extended to handle functions which depend in a limited way on a number of independent random variables. This ‘method of bounded differences’ has over the last dozen or so years had a great impact in probabilistic methods in discrete mathematics and in the mathematics of operational research and theoretical computer science. Recently Talagrand introduced an exciting new method for bounding probabilities of large deviations, which often proves superior to the bounded differences approach. In this chapter we introduce and survey these two approaches and some of their applications.

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References

  1. Ahlswede R., Gâcs P. and Körner J. (1976): Bounds on conditional probabilities with applications in multi-user communication, Z. Wahrscheinlichkeitstheorie verw. Geb. 34, 157–177. (Erratum (1977) 39, 353–354.)

    Google Scholar 

  2. Alon N., Kim J.H. and Spencer J. (1997): Nearly perfect matchings in regular simple hypergraphs, Israel J. Math. 100, 171–188.

    MathSciNet  MATH  Google Scholar 

  3. Alon N. and Spencer N. (1992): The Probabilistic Method, John Wiley Sons.

    Google Scholar 

  4. Angluin D. and Valiant L. (1979): Fast probabilistic algorithms for Hamiltonian circuits and matchings, J. Computer and System Sciences 18, 155–193.

    Article  MathSciNet  MATH  Google Scholar 

  5. Avram F. and Bertsimas D. (1992): The minimum spanning tree constant in geometrical probability and under the independent model: a unified approach, Annals of Applied Probability 2, 113–130.

    Article  MathSciNet  MATH  Google Scholar 

  6. Azuma K. (1967): Weighted sums of certain dependent random variables, Tôkuku Math. J. 19, 357–367.

    MathSciNet  MATH  Google Scholar 

  7. Bennett G. (1962): Probability inequalities for the sum of independent random variables, J. Amer. Statist. Assoc. 57, 33–45.

    Article  MATH  Google Scholar 

  8. Beveridge A., Frieze A. and McDiarmid C. (1998): Random minimum length spanning trees in regular graphs, Combinatorica, to appear.

    Google Scholar 

  9. Bollobâs B. (1985): Random Graphs, Academic Press.

    Google Scholar 

  10. Bollobfís B. (1997): Martingales, isoperimetric inequalities and random graphs, Colloq. Math. Soc. Janos Bolyai 52, 113–139.

    Google Scholar 

  11. Bollobâs B. (1988): The chromatic number of random graphs, Combinatorica 8, 49–55.

    Article  MathSciNet  MATH  Google Scholar 

  12. Bollobâs B. (1990): Sharp concentration of measure phenomena in the theory of random graphs, in Random Graphs ‘87, (M. Karonski, J. Jaworski and A. Rucidski, editors ), John Wiley and Sons, 1–15.

    Google Scholar 

  13. Bollobâs B. and Brightwell G. (1992): The height of a random partial order: concentration of measure, Ann. Appl. Probab. 2, 1009–1018.

    Article  MathSciNet  MATH  Google Scholar 

  14. Chernoff H. (1952): A measure of the asymptotic efficiency for tests of a hypothesis based on the sum of observables, Ann. Math. Statist. 23, 493–509.

    Article  MathSciNet  MATH  Google Scholar 

  15. Chvâtal V. (1979): The tail of the hypergeometric distribution, Discrete Mathematics 25, 285–287.

    Article  MathSciNet  MATH  Google Scholar 

  16. Coffman E.G. and Lueker G.S. (1991): Probabilistic Analysis of Packing and Partitioning Algorithms, Wiley, New York.

    Google Scholar 

  17. Csiszâr I. and Körner J. (1981): Information Theory: Coding Theorems for Discrete Memoryless Systems, Academic Press, New York.

    MATH  Google Scholar 

  18. Dembo A. (1997): Information inequalities and concentration of measure, Ann. Probab. 25, 927–939.

    Article  MathSciNet  MATH  Google Scholar 

  19. Dembo A. and Zeitouni O. (1993): Large Deviation Techniques, Jones and Bartlett.

    Google Scholar 

  20. Durrett R. (1996): Probability: Theory and Examples, Second edition, Duxbury Press.

    Google Scholar 

  21. Freedman D.A. (1975): On tail probabilities for martingales, Ann. Probab. 3, 100–118.

    Article  MATH  Google Scholar 

  22. Feller W.J. (1968): An Introduction to Probability Theory and its Applications, Volume 1, Third Edition, John Wiley Sons, New York.

    MATH  Google Scholar 

  23. Frieze A.M. (1985): On the value of a random minimum spanning tree problem, Discrete Applied Mathematics 10, 47–56.

    Article  MathSciNet  MATH  Google Scholar 

  24. Frieze A.M. and McDiarmid C.J.H. (1989): On random minimum length spanning trees, Combinatorica 9, 363–374.

    Article  MathSciNet  MATH  Google Scholar 

  25. Frieze A.M. and McDiarmid C.J.H. (1997): Algorithmic theory of random graphs, Random Structures and Algorithms 10, 5–42.

    Article  MathSciNet  Google Scholar 

  26. Grable D.A. (1998): A large deviation inequality for functions of independent multi-way choices, Combinatorics, Probability and Computing 7, 57–63.

    Article  MathSciNet  MATH  Google Scholar 

  27. Grable D.A. and Panconesi A. (1997): Nearly optimal distributed edge colouring in O(log log n) rounds, Random Structures and Algorithms 10, 385–405.

    Article  MathSciNet  MATH  Google Scholar 

  28. Grimmett G.R. and Stirzaker D.R. (1992): Probability and Random Processes, Second edition, Oxford University Press.

    Google Scholar 

  29. Hoeffding W.J. (1963): Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58, 713–721.

    Article  MathSciNet  Google Scholar 

  30. Janson S. (1995): The minimal spanning tree in a complete graph and a functional limit theorem for trees in a random graph, Random Structures and Algorithms 7, 337–355.

    Article  MathSciNet  MATH  Google Scholar 

  31. Johnson W. and Schechtman G. (1991): Remarks on Talagrand’s deviation inequality for Rademacher’s functions, Lecture Notes in Mathematics 1470, Springer-Verlag, 72–77.

    Google Scholar 

  32. Kahn J. (1996): Asymptotically good list colorings, J. Combinatorial Theory A 73, 1–59.

    Article  MathSciNet  MATH  Google Scholar 

  33. Kamath A., Motwani R., Palem K. and Spirakis P. (1995): Tail bounds for occupancy and the satisfiability threshold conjecture, Random Structures and Algorithms 7, 59–80.

    Article  MathSciNet  MATH  Google Scholar 

  34. Kim J.H. (1995): On Brooks’ theorem for sparse graphs, Combinatorics, Probability and Computing 4, 97–132.

    Article  MATH  Google Scholar 

  35. Kim J.H. (1995): The Ramsey number R(3, t) has order of magnitude t2/ log t, Random Structures and Algorithms 7, 173–207.

    Article  MathSciNet  MATH  Google Scholar 

  36. Knuth D.E. (1973): The Art of Computer Programming Volume 3: Sorting and Searching, Addison-Wesley.

    Google Scholar 

  37. Leader I. (1991): Discrete isoperimetric inequalities, Proc. Sympos. Appl. Math. 44, 57–80.

    MathSciNet  Google Scholar 

  38. Ledoux M. and Talagrand M. (1991): Probability in Banach Spaces, Springer-Verlag.

    Google Scholar 

  39. Lipster R. Sh. and Shiryaev A.N. (1989): Theory of Martingales Kluwer, Dordrecht.

    Google Scholar 

  40. Marton K. (1986): A simple proof of the blowing-up lemma, IEEE Transactions in Information Theory, 32, 445–446.

    Article  MathSciNet  MATH  Google Scholar 

  41. Marton K. (1996): Bounding d-distance by informational divergence: a method to prove measure concentration, Ann. Probab. 24, 857–866.

    Article  MathSciNet  MATH  Google Scholar 

  42. Marton K. (1996): A measure concentration inequality for contracting Markov chains, Geometric and Functional Analysis 6 556–571. (Erratum (1997) 7, 609–613.)

    Google Scholar 

  43. Marton K. and Shields P.C. (1994): The positive divergence and blowing-up properties, Israel J. Math. 86, 331–348.

    MathSciNet  MATH  Google Scholar 

  44. Maurey B. (1979): Construction de suites symétriques, Compt. Rend. Acad. Sci. Paris 288, 679–681.

    MathSciNet  MATH  Google Scholar 

  45. McDiarmid C. (1989): On the method of bounded differences, in Surveys in Combinatorics, ed J. Siemons, London Mathematical Society Lecture Note Series 141, Cambridge University Press.

    Google Scholar 

  46. McDiarmid C. (1990): On the chromatic number of random graphs, Random Structures and Algorithms 1, 435–442.

    Article  MathSciNet  MATH  Google Scholar 

  47. McDiarmid C. (1997): Centering sequences with bounded differences, Combinatorics, Probability and Computing 6, 79–86.

    Article  MathSciNet  MATH  Google Scholar 

  48. McDiarmid C. (1998): Concentration for minimum spanning tree lengths, manuscript.

    Google Scholar 

  49. McDiarmid C. and Hayward R. (1996): Large deviations for quicksort, J. Algorithms 21, 476–507.

    Article  MathSciNet  MATH  Google Scholar 

  50. Milman V. and Schechtman G. (1986): Asymptotic theory of finite dimensional normed spaces, Lecture Notes in Math. 1200, Springer-Verlag.

    Google Scholar 

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

    Google Scholar 

  52. Penrose M. (1998): Random minimum spanning tree and percolation on the n-cube, Random Structures and Algorithms 12, 369–382.

    Article  MathSciNet  Google Scholar 

  53. Platzman L.K. and Bartholdi J.J. (1989): Spacefilling curves and the planar traveling salesman problem, J. Assoc. Comput. Mach 36, 719–737.

    Article  MathSciNet  MATH  Google Scholar 

  54. Rhee W.T. and Talagrand M. (1987): Martingale inequalities and NP-complete problems, Math. Oper. Res. 12, 177–181.

    Article  MathSciNet  MATH  Google Scholar 

  55. Rhee W.T. and Talagrand M. (1989): Martingale inequalities, interpolation and NP-complete problems, Math. Oper. Res. 14, 189–202.

    Article  MathSciNet  MATH  Google Scholar 

  56. Rhee W.T. and Talagrand M. (1989): A sharp deviation for the stochastic traveling salesman problem, Ann. Probab. 17, 1–8.

    Article  MathSciNet  MATH  Google Scholar 

  57. Ross S.M. (1996): Stochastic Processes, Second edition, Wiley.

    MATH  Google Scholar 

  58. Schmidt J., Siegel A. and Srinivasan A. (1995): Chernoff-Hoeffding bounds for applications with limited independence, SIAM J. Discrete Math. 8, 223–250.

    Article  MathSciNet  MATH  Google Scholar 

  59. Sedgewick R. and Flajolet P. (1996): Analysis of Algorithms, Addison-Wesley.

    Google Scholar 

  60. Shamir E. and Spencer J. (1987): Sharp concentration of the chromatic number on random graphs G„,,, Combinatorica 7, 374–384.

    Article  MathSciNet  Google Scholar 

  61. Shiryaev A.N. (1996): Probability, Second edition, Graduate Texts in Mathematics 95, Springer.

    Google Scholar 

  62. Steele J.M. (1995): Variations on the long increasing subsequence theme of ErdHos and Szekeres, in Discrete Probability and Algorithms, D. Aldous, P. Dia conis and J.M. Steele, eds., Volumes in Mathematics and its Applications 72, Springer-Verlag, New York, 111–131.

    Google Scholar 

  63. Steele J.M. (1997): Probability Theory and Combinatorial Optimization, SIAM CBMS 69.

    Google Scholar 

  64. Steiger W.L. (1967): Some Kolmogoroff-type inequalities for bounded random variables, Biometrika 54, 641–647.

    MathSciNet  MATH  Google Scholar 

  65. Steiger W.L. (1969): A best possible Kolmogoroff-type inequality for martingales and a characteristic property, Ann. Math. Statist. 40, 764–769.

    Article  MathSciNet  MATH  Google Scholar 

  66. Steiger W.L. (1970): Bernstein’s inequality for martingales, Z. Wahrscheinlichkeitstheorie verw. Geb. 16, 104–106.

    Article  MathSciNet  MATH  Google Scholar 

  67. Talagrand M. (1991): A new isoperimetric inequality for product measure and the tails of sums of independent random variables, Geometric and Functional Analysis 1, 211–223.

    Article  MathSciNet  MATH  Google Scholar 

  68. Talagrand M. (1995): Concentration of measure and isoperimetric inequalities in product spaces, Publ. Math. Institut des Hautes Études Scientifiques 81, 73–205.

    Article  MathSciNet  MATH  Google Scholar 

  69. Talagrand M. (1996): A new look at independence, Annals of Probability 24, 1 31.

    MathSciNet  Google Scholar 

  70. Talagrand M. (1996): New concentration inequalities in product spaces, Invent. Math. 126, 505–563.

    MathSciNet  MATH  Google Scholar 

  71. Talagrand M. (1996): Transportation cost for Gaussian and other product measures, Geometric and Functional Analysis 6, 587–600.

    Article  MathSciNet  MATH  Google Scholar 

  72. Williams D (1991): Probability with Martingales, Cambridge University Press.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Statistics, University of Oxford, UK

    Colin McDiarmid

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  1. Colin McDiarmid
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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  & 

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McDiarmid, C. (1998). Concentration. 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_6

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  • DOI: https://doi.org/10.1007/978-3-662-12788-9_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08426-3

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