Hostname: page-component-5db58dd55d-h5th4 Total loading time: 0 Render date: 2026-06-01T19:39:38.471Z Has data issue: false hasContentIssue false
  • English
  • Français

Probabilistic Divide-and-Conquer: A New Exact Simulation Method, With Integer Partitions as an Example

Part of: Combinatorics Combinatorial probability

Published online by Cambridge University Press:  22 January 2016

RICHARD ARRATIA  and
STEPHEN DeSALVO
RICHARD ARRATIA
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA (e-mail: rarratia@math.usc.edu)
STEPHEN DeSALVO
Affiliation:
Department of Mathematics, University of California Los Angeles, Los Angeles, CA 90024, USA (e-mail: stephendesalvo@math.ucla.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose a new method, probabilistic divide-and-conquer, for improving the success probability in rejection sampling. For the example of integer partitions, there is an ideal recursive scheme which improves the rejection cost from asymptotically order n 3/4 to a constant. We show other examples for which a non-recursive, one-time application of probabilistic divide-and-conquer removes a substantial fraction of the rejection sampling cost.

We also present a variation of probabilistic divide-and-conquer for generating i.i.d. samples that exploits features of the coupon collector's problem, in order to obtain a cost that is sublinear in the number of samples.

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05A17: Partitions of integers

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 3 , May 2016 , pp. 324 - 351
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Almkvist, G. (1991) Exact asymptotic formulas for the coefficients of nonmodular functions. J. Number Theory 38 145–160.CrossRefGoogle Scholar
[2] Almkvist, G. and Andrews, G. E. (1991) A Hardy–Ramanujan formula for restricted partitions. J. Number Theory 38 135–144.CrossRefGoogle Scholar
[3] Alonso, L. (1994) Uniform generation of a Motzkin word. Theoret. Comput. Sci. 134 529–536.CrossRefGoogle Scholar
[4] Andrews, G. E. (1984) The Theory of Partitions, Cambridge Mathematical Library.CrossRefGoogle Scholar
[5] Arratia, R. (2002) On the amount of dependence in the prime factorization of a uniform random integer. In Contemporary Combinatorics, Vol. 10 of Bolyai Society Mathematical Studies, János Bolyai Mathematical Society, pp. 29–91.Google Scholar
[6] Arratia, R., Gordon, L. and Waterman, M. S. (1990) The Erdős–Rényi law in distribution, for coin tossing and sequence matching. Ann. Statist. 18 529–570.CrossRefGoogle Scholar
[7] Arratia, R. and Tavaré, S. (1994) Independent process approximations for random combinatorial structures. Adv. Math. 104 90–154.CrossRefGoogle Scholar
[8] Arratia, R. and Waterman, M. S. (1985) Critical phenomena in sequence matching. Ann. Probab. 13 1236–1249.CrossRefGoogle Scholar
[9] Bodini, O., Fusy, E. and Pivoteau, C. (2010) Random sampling of plane partitions. Combin. Probab. Comput. 19 201–226.CrossRefGoogle Scholar
[10] Bollobás, B. (2001) Random Graphs, second edition, Vol. 73 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.CrossRefGoogle Scholar
[11] Cover, T. M. and Thomas, J. A. (1991) Elements of Information Theory, Wiley Series in Telecommunications, Wiley.Google Scholar
[12] Denise, A. and Zimmermann, P. (1999) Uniform random generation of decomposable structures using floating-point arithmetic. Theoret. Comput. Sci. 218 233–248.CrossRefGoogle Scholar
[13] DeSalvo, S. (2012) Probabilistic divide-and-conquer: a new exact simulation method, with integer partitions as an example and lower bound expansions for random Bernoulli matrices via novel integer partitions. PhD thesis, University of Southern California.Google Scholar
[14] DeSalvo, S. (2014) Probabilistic divide-and-conquer: deterministic second half. arXiv:1411.6698 Google Scholar
[15] DeSalvo, S. and Pak, I. (2015) Log-concavity of the partition function. Ramanujan J. 38 61–73.CrossRefGoogle Scholar
[16] Devroye, L. (1986) Non-Uniform Random Variate Generation, Springer.CrossRefGoogle Scholar
[17] Diffie, W. and Hellman, M. E. (1977) Special feature exhaustive cryptanalysis of the NBS data encryption standard. Computer 10 74–84.CrossRefGoogle Scholar
[18] Duchon, P., Flajolet, P., Louchard, G. and Schaeffer, G. (2004) Boltzmann samplers for the random generation of combinatorial structures. Combin. Probab. Comput. 13 577–625.CrossRefGoogle Scholar
[19] Engel, B. (2014) Log-concavity of the overpartition function. arXiv: 1412.4603Google Scholar
[20] Erdős, P. and Lehner, J. (1941) The distribution of the number of summands in the partitions of a positive integer. Duke Math. J. 8 335–345.CrossRefGoogle Scholar
[21] Fristedt, B. (1993) The structure of random partitions of large integers. Trans. Amer. Math. Soc. 337 703–735.CrossRefGoogle Scholar
[22] Hardy, G. H. and Ramanujan, S. (1918) Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. s2–17 (1) 75–115.CrossRefGoogle Scholar
[23] Johansson, F. (2012) Efficient implementation of the Hardy–Ramanujan–Rademacher formula. LMS J. Comput. Math. 15 341–359.CrossRefGoogle Scholar
[24] Knessl, C. and Keller, J. B. (1990) Partition asymptotics from recursion equations. SIAM J. Appl. Math. 50 323–338.CrossRefGoogle Scholar
[25] Knuth, D. E. and Yao, A. C. (1976) The complexity of nonuniform random number generation. In Algorithms and complexity: Proc. Sympos., Carnegie-Mellon University, Pittsburgh, PA, 1976, Academic Press, pp. 357–428.Google Scholar
[26] Lam, T., Lapointe, L., Morse, J. and Shimozono, M. (2010) Affine insertion and Pieri rules for the affine Grassmannian. Mem. Amer. Math. Soc. 208 (977).Google Scholar
[27] Lehmer, D. H. (1938) On the series for the partition function. Trans. Amer. Math. Soc. 43 271–295.CrossRefGoogle Scholar
[28] Lehmer, D. H. (1939) On the remainders and convergence of the series for the partition function. Trans. Amer. Math. Soc. 46 362–373.CrossRefGoogle Scholar
[29] Mathematica (2010) Mathematica Edition: Version 8.0, Wolfram Research, Inc. Google Scholar
[30] MATLAB (2011) Version 7.12.0.635 (R2011a), The MathWorks, Inc. Google Scholar
[31] McKay, B. D. and Wormald, N. C. (1990) Uniform generation of random regular graphs of moderate degree. J. Algorithms 11 52–67.CrossRefGoogle Scholar
[32] McLaughlin, J. and Parsell, S. (2012) A Hardy–Ramanujan–Rademacher-type formula for (r, s)-regular partitions. Ramanujan J. 28 253–271.CrossRefGoogle Scholar
[33] Nicolas, J.-L. (1978) Sur les entiers N pour lesquels il y a beaucoup de groupes abéliens d'ordre N . Ann. Inst. Fourier (Grenoble) 28 1–16.CrossRefGoogle Scholar
[34] Nijenhuis, A. and Wilf, H. S. (1975) A method and two algorithms on the theory of partitions. J. Combin. Theory Ser. A 18 219–222.CrossRefGoogle Scholar
[35] Nijenhuis, A. and Wilf, H. S. (1978) Combinatorial Algorithms: For Computers and Calculators, Academic Press.Google Scholar
[36] Pak, I. (2006) Partition bijections: A survey. Ramanujan J. 12 5–75.CrossRefGoogle Scholar
[37] Pittel, B. (1997) On a likely shape of the random Ferrers diagram. Adv. Appl. Math. 18 432–488.CrossRefGoogle Scholar
[38] Pittel, B. (1997) Random set partitions: asymptotics of subset counts. J. Combin. Theory Ser. A 79 326–359.CrossRefGoogle Scholar
[39] Rademacher, H. (1937) A convergent series for the partition function p(n). Proc. Nat. Acad. Sci. 23 78–84.CrossRefGoogle ScholarPubMed
[40] Remmel, J. B. (1982) Bijective proofs of some classical partition identities. J. Combin. Theory Ser. A 33 273–286.CrossRefGoogle Scholar
[41] Sills, A. V. (2010) Rademacher-type formulas for restricted partition and overpartition functions. Ramanujan J. 23 253–264.CrossRefGoogle Scholar
[42] von Neumann, J. (1951) Various techniques used in connection with random digits. Journal of Research of the National Bureau of Standards, Appl. Math. Series 3, 36–38.Google Scholar