Simulating size-constrained galton-watson trees

SIAM Journal on Computing

Volumn 41, Issue 1, 2012, Pages 1-11

  Devroye, Luc ^a  

^a MCGILL UNIVERSITY   (Canada)

Author keywords

Catalan trees; Cayley trees; Expected time analysis; Galton Watson branching process; Random trees; Random variate generation; Simulation

Indexed keywords

CATALANS; CAYLEY TREES; EXPECTED TIME; GALTON-WATSON BRANCHING PROCESS; RANDOM TREE; RANDOM VARIATES; SIMULATION;

FORESTRY;

TREES (MATHEMATICS);

ALGORITHMS; RANDOM PROCESSES; SIMULATION;

EID: 84861589634     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/090766632     Document Type: Article

References (64)

1. J. H. Ahrens and U. Dieter, Computer methods for sampling from gamma, beta, Poisson and binomial distributions, Computing, 12 (1974), pp. 223-246.
2. J. H. Ahrens and U. Dieter, Sampling from binomial and Poisson distributions: A method with bounded computation times, Computing, 25 (1980), pp. 193-208.
3. D. Aldous, The random walk construction of uniform spanning trees and uniform labelled trees, SIAM J. Discrete Math., 3 (1990), pp. 450-465.
4. D. Aldous, Asymptotic fringe distributions for general families of random trees, Ann. Appl. Probab., 1 (1991), pp. 228-266.
5. L. Alonso, Structures arbortescentes, algorithmes de géné ration, probléme d'inclusion, relations maximin, Ph.D. thesis, Université Paris-Sud, Orsay, 1992.
6. L. Alonso, J. L. Rémy, and R. Schott, Uniform generation of a Schröder tree, Inform. Process. Lett., 64 (1997), pp. 305-308.
7. L. Alonso and R. Schott, Random Generation of Trees, Kluwer, Boston, 1995.
8. L. Alonso, J. L. Rémy, and R. Schott, A linear-Time algorithm for the generation of trees, Algorithmica, 17 (1997), pp. 162-182.
9. D. B. Arnold and M. R. Sleep, Uniform random number generation of n balanced parenthesis strings, ACM Trans. Programming Languages Systems, 2 (1980), pp. 122-128.
10. K. B. Athreya and P. E. Ney, Branching Processes, Springer-Verlag, Berlin, 1972.
11. C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy, and D. Gouyou-Beauchamps, Generating functions for generating trees, Discrete Math., 246 (2002), pp. 29-55.
12. E. Barcucci, A. Del Lungo, and E. Pergola, Random generation of trees and other combinatorial objects, Theoret. Comput. Sci., 218 (1999), pp. 219-232.
13. E. Barcucci, R. Pinzani, and R. Sprugnoli, The random generation of directed animals, Theoret. Comput. Sci., 127 (1992), pp. 333-350.
14. A. Cayley, A theorem on trees, Quart. J. Math., 23 (1889), pp. 376-378.
15. L. Chottin and R. Cori, Une preuve combinatoire de la rationalité d'une série génératrice associée aux arbres, RAIRO Inform. Théor., 16 (1982), pp. 113-128.
16. L. Comtet, Advanced Combinatorics, D. Reidel, Dordrecht, 1994.
17. N. G. de Bruijn, D. E. Knuth, and S. O. Rice, The average height of planted plane trees, in Graph Theory and Computing, R. C. Read, ed., Academic Press, NY, 1972, pp. 15-22.
18. A. Denise, Méthodes de génération aléatoire d'objets combinatoires de grande taille et problémes d'é numération, Thése, Université Bordeaux I, France, 1994.
19. N. Dershowitz and S. Zaks, The cycle lemma and some applications, European J. Combin., 11 (1990), pp. 35-40.
20. L. Devroye, Non-Uniform Random Variate Generation, Springer-Verlag, New York, 1986.
21. L. Devroye, A simple generator for discrete log-concave distributions, Computing, 39 (1987), pp. 87-91.
22. L. Devroye, Generating sums in constant average time, in Proceedings of the 1988 Winter Simulation Conference, M. A. Abrams, P. L. Haigh, and J. C. Comfort, eds., IEEE, San Diego, CA, 1988, pp. 425-431.
23. P. Duchon, P. Flajolet, G. Louchard, and G. Schaeffer, Random Sampling from Boltzmann Principles, Tech. Rep. ALCOMFT-TR-01-189, INRIA, Paris, 2001.
24. P. Duchon, P. Flajolet, G. Louchard, and G. Schaeffer, Random sampling from Boltzmann principles, in Automata, Languages and Programming (ICALP 2002), P. Widmayer, F. Triguero, R. Morales, M. Hennessy, S. Eidenbenz, and R. Conejo, eds., Lecture Notes in Comput. Sci. 2380, Springer-Verlag, New York, 2002, pp. 501-513.
25. P. Duchon, P. Flajolet, G. Louchard, and G. Schaeffer, Boltzmann samplers for the random generation of combinatorial structures, Combin. Probab. Comput., 13 (2004), pp. 577-625.
26. M. Dwass, The total progeny in a branching process, J. Appl. Probab., 6 (1969), pp. 682-686.
27. P. Flajolet, P. Zimmerman, and B. van Cutsem, A calculus for the random generation of labelled combinatorial structures, Theoret. Comput. Sci., 132 (1994), pp. 1-35.
28. D. Gouyou-Beauchamps, Quelques exemples d'algorithmes de génération aléatoire, 1993.
29. D. Gouyou-Beauchamps, Combinatorics and random generation, in Algorithms Seminar 2001-2002, F. Chyzak, ed., INRIA, Paris, 2003, pp. 177-182.
30. C. C. Heyde and E. Seneta, I.J. Bienaymé: Statistical Theory Anticipated, Studies in the History of Mathematics and Physical Sciences 3, Springer-Verlag, New York, Heidelberg, Berlin, 1977.
31. T. Hickey and J. Cohen, Uniform random generation of strings in a context-free language, SIAM J. Comput., 12 (1983), pp. 645-655.
32. W. Hörmann, The generation of binomial random variates, J. Statist. Comput. Simul., 46 (1993), pp. 101-110.
33. W. Hörmann, J. Leydold, and G. Derflinger, Automatic Nonuniform Random Variate Generation, Springer-Verlag, Berlin, 2004.
34. S. Janson, T. Luczak, and A. Rucinski, Random Graphs, Wiley-Interscience, New York, 2000.
35. V. Kachitvichyanukul and B. W. Schmeiser, Binomial random variate generation, Comm. ACM, 31 (1988), pp. 216-222.
36. V. Kachitvichyanukul and B. W. Schmeiser, Algorithm 678: BTPEC: Sampling from the binomial distribution, ACM Trans. Math. Software, 15 (1989), pp. 394-397.
37. D. G. Kendall, The genealogy of branching processes before (and after) 1873, Bull. London Math. Soc., 7 (1975), pp. 225-254.
38. D. P. Kennedy, The Galton-Watson process conditioned on the total progeny, J. Appl. Probab., 12 (1975), pp. 800-806.
39. P. Klingsberg, Doctorial Dissertation, University of Washington, Seattle, WA, 1977.
40. D. E. Knuth, The Art of Computer Programming, Vol. 2, 2nd ed., Addison-Wesley, Reading, MA, 1981.
41. D. E. Knuth, The Art of Computer Programming. Vol. 1, Fundamental Algorithms, 3rd ed., Addison-Wesley, Reading, MA, 1997.
42. V. F. Kolchin, Random Mappings, Optimization Software Inc., New York, 1986.
43. M. Lothaire, Combinatorics on Words, Encyclopedia Math. Appl. 17, Addison-Wesley, 1983.
44. M. Luczak and P. Winkler, Building uniformly random subtrees, Random Structures Algorithms, 27 (2004), pp. 420-443.
45. E. Mäkinen, Generating random binary trees-A survey, Inform. Sci., 115 (1999), pp. 123-136.
46. E. Mäkinen and J. Siltaneva, A Note on Rémy's Algorithm for Generating Random Binary Trees, Technical report, Department of Computer and Information Sciences, University of Tampere, Finland, 1999.
47. U. Manber, Introduction to Algorithms: A Creative Approach, Addison-Wesley, Reading, MA, 1989.
48. A. Meir and J. W. Moon, The distance between points in random trees, J. Combin. Theory, 8 (1970), pp. 99-103.
49. A. Meir and J. W. Moon, On the altitude of nodes in random trees, Canad. J. Math., 30 (1978), pp. 997-1015.
50. J. W. Moon, Counting Labelled Trees, Canadian Mathematical Congress, Montreal, 1970.
51. A. Nijenhuis and H. S. Wilf, Combinatorial Algorithms, 2nd ed., Academic Press, New York, 1978.
52. Yu. L. Pavlov, Random Forests, VSP, Utrecht, 2000.
53. V. V. Petrov, Sums of Independent Random Variables, Springer-Verlag, Berlin, 1975.
54. V. V. Petrov, Limit Theorems of Probability Theory, Oxford Science Publications, Clarendon Press, Oxford, UK, 1995.
55. A. Prüfer, Neuer Beweis eines Satzes über Permutationen, Archiv Math. Phys., 3 (1918), pp. 142-144.
56. J. L. Rémy, Un procédé itératif de dénombrement d'arbres binaires et son application à leur génération aléatoire, RAIRO Theoret. Inform. Appl., 19 (1985), pp. 179-195.
57. C. Savage, A survey of combinatorial Gray codes, SIAM Rev., 39 (1997), pp. 605-629.
58. B. W. Schmeiser and A. J. G. Babu, Beta variate generation via exponential majorizing functions, Oper. Res., 28 (1980), pp. 917-926.
59. J. Siltaneva, Random Generation of Binary Trees, Master's Thesis, Department of Computer and Information Sciences, University of Tampere, Finland, 2000 (in Finnish).
60. E. Stadlober, Sampling from Poisson, Binomial and Hypergeometric Distributions: Ratio of Uniforms as a Simple Fast Alternative, Habilitationsschrift, Institute of Statistics, Technical University of Graz, Austria, 1988.
61. E. Stadlober, Binomial random variate generation: A method based on ratio of uniforms, Amer. J. Math. Management Sci., 9 (1989), pp. 1-20.
62. E. Stadlober, The ratio of uniforms approach for generating discrete random variates, J. Comput. Appl. Math., 31 (1990), pp. 181-189.
63. J. S. Vitter and P. Flajolet, Analysis of algorithms and data structures, in Handbook of Theoretical Computer Science, J. van Leeuwen, ed., North-Holland, Amsterdam, 1990, pp. 431-524.
64. H. S. Wilf, Combinatorial Algorithms: An Update, SIAM, CBMS-NSF Reg. Conf. Ser. App. Math. 55, Philadelphia, 1989.