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An Analysis of Budgeted Parallel Search on Conditional Galton–Watson Trees

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Abstract

Recently Avis and Jordan have demonstrated the efficiency of a simple technique called budgeting for the parallelization of a number of tree search algorithms. The idea is to limit the amount of work that a processor performs before it terminates its search and returns any unexplored nodes to a master process. This limit is set by a critical budget parameter which determines the overhead of the process. In this paper we study the behaviour of the budget parameter on conditional Galton–Watson trees obtaining asymptotically tight bounds on this overhead. We present empirical results to show that this bound is surprisingly accurate in practice.

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Notes

  1. http://cgm.cs.mcgill.ca/~avis/doc/tutorial.html.

  2. All computational results in the paper were obtained on mai20 at Kyoto University: 2x Xeon E5-2690 (10-core 3.0GHz), 20 cores, 128GB memory, 3TB hard drive.

References

  1. Aldous, D.: The continuum random tree. II. An overview. In: Stochastic Analysis (Durham, 1990), vol. 167, pp. 23–70, Cambridge University Press, Cambridge (1991)

    Chapter  Google Scholar 

  2. Aldous, D.: The continuum random tree. I. Ann. Probab. 19, 1–28 (1991)

    Article  MathSciNet  Google Scholar 

  3. Aldous, D.: Asymptotic fringe distributions for general families of random trees. Ann. Appl. Probab. 1, 228–266 (1991)

    Article  MathSciNet  Google Scholar 

  4. Aldous, D.: The continuum random tree. III. Ann. Probab. 21, 248–289 (1993)

    Article  MathSciNet  Google Scholar 

  5. Athreya, K.B., Ney, P.E.: Branching Processes. Springer, Berlin (1972)

    Book  Google Scholar 

  6. Avis, D., Jordan, C.: mplrs: a scaleable parallel vertex/facet enumeration code. arXiv:1511.06487 (2015)

  7. Avis, D., Jordan, C.: A parallel framework for reverse search using mts. arXiv:1610.07735 (2016)

  8. Bennies, J., Kersting, G.: A random walk approach to Galton–Watson trees. J. Theor. Probab. 13, 777–803 (2000)

    Article  MathSciNet  Google Scholar 

  9. Blumofe, N., Leiserson, C.: Scheduling multithreaded computations by work stealing. J. ACM 46, 720–748 (1999)

    Article  MathSciNet  Google Scholar 

  10. Duquesne, T.: A limit theorem for the contour process of conditioned Galton–Watson trees. Ann. Probab. 31, 996–1027 (2003)

    Article  MathSciNet  Google Scholar 

  11. Dwass, M.: The total progeny in a branching process. J. Appl. Probab. 6, 682–686 (1969)

    Article  Google Scholar 

  12. Flajolet, P., Odlyzko, A.: The average height of binary trees and other simple trees. J. Comput. Syst. Sci. 25, 171–213 (1982)

    Article  MathSciNet  Google Scholar 

  13. Hooker, J.N.: Testing heuristics: we have it all wrong. J. Heuristics 1, 33–42 (1995)

    Article  Google Scholar 

  14. Ito, K., McKean, H.P.: Diffusion Processes and Their Sample Paths. Springer, Berlin (1974)

    MATH  Google Scholar 

  15. Janson, S.: Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation. Probab. Surv. 9, 103–252 (2012)

    Article  MathSciNet  Google Scholar 

  16. Kennedy, D.P.: The Galton–Watson process conditioned on the total progeny. J. Appl. Probab. 12, 800–806 (1975)

    Article  MathSciNet  Google Scholar 

  17. Kennedy, D.P.: The distribution of the maximum Brownian excursion. J. Appl. Probab. 13, 371–376 (1976)

    Article  MathSciNet  Google Scholar 

  18. Kolchin, V.F.: Branching processes and random trees.In: Problems in Cybernetics, Combinatorial Analysis and Graph Theory (in Russian), pp. 85–97, Nauka, Moscow (1980)

  19. Kolchin, V.F.: Random Mappings. Optimization Software Inc., New York (1986)

    MATH  Google Scholar 

  20. Le Gall, J.-F.: Marches aléatoires, mouvement Brownien etprocessus de branchement.In: Séminaire de ProbabilitésXXIII, edited by Azéma, J., Meyer, P.A., Yor, M. vol. 1372, pp. 258–274. Lecture Notes in Mathematics, Springer-Verlag, Berlin (1989)

  21. Le Gall, J.-F.: Random trees and applications. Probab. Surv. 2, 245–311 (2005)

    Article  MathSciNet  Google Scholar 

  22. Marckert, J.F., Mokkadem, A.: The depth first processes of Galton–Watson trees converge to the same Brownian excursion. Ann. Probab. 31, 1655–1678 (2003)

    Article  MathSciNet  Google Scholar 

  23. Mattson, T., Saunders, B., Massingill, B.: Patterns for Parallel Programming. Addison-Wesley, Boston (2004)

    Google Scholar 

  24. McCreesh, C., Prosser, P.: The shape of the search tree for the maximum clique problem and the implications for parallel branch and bound. ACM Trans. Parallel Process. 2, 8:1–8:27 (2015)

    MATH  Google Scholar 

  25. Meir, A., Moon, J.W.: On the altitude of nodes in random trees. Can. J. Math. 30, 997–1015 (1978)

    Article  MathSciNet  Google Scholar 

  26. Moon, J.W.: Counting Labelled Trees. Canadian Mathematical Congress, Montreal (1970)

    MATH  Google Scholar 

  27. Petrov, V.V.: Sums of Independent Random Variables. Springer, Berlin (1975)

    Book  Google Scholar 

  28. Rényi, A., Szekeres, G.: On the height of trees. J. Aust. Math. Soc. 7, 497–507 (1967)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We gratefully acknowledge helpful conversations with Louigi Addario-Berry and Charles Jordan. The research of Avis was supported by a JSPS Grant No. (24300002) Kakenhi Grant and a Grant-in-Aid for Scientific Research on Innovative Areas, ‘Exploring the Limits of Computation (ELC)’. The research of Devroye was supported by the Natural Sciences and Engineering Research Council of Canada.

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Authors and Affiliations

  1. School of Informatics, Kyoto University, Kyoto, Japan

    David Avis

  2. School of Computer Science and GERAD, McGill University, Montreal, Quebec, Canada

    David Avis

  3. School of Computer Science, McGill University, Montreal, Quebec, Canada

    Luc Devroye

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  1. David Avis
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  2. Luc Devroye
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Correspondence to David Avis.

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Avis, D., Devroye, L. An Analysis of Budgeted Parallel Search on Conditional Galton–Watson Trees. Algorithmica 82, 1329–1345 (2020). https://doi.org/10.1007/s00453-019-00645-x

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