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MCTS Experiments on the Voronoi Game

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7168)

Abstract

Monte-Carlo Tree Search (MCTS) is a powerful tool in games with a finite branching factor. The paper describes an artificial player playing the Voronoi game, a game with an infinite branching factor. First, it shows how to use MCTS on a discretization of the Voronoi game, and the effects of enhancements such as RAVE and Gaussian processes (GP). Then a set of experimental results shows that MCTS with UCB+RAVE or with UCB+GP are good first solutions for playing the Voronoi game without domain-dependent knowledge. Moreover, the paper shows how the playing level can be greatly improved by using geometrical knowledge about Voronoi diagrams. The balance of diagrams is the key concept. A new set of experimental results shows that a player using MCTS and geometrical knowledge outperforms a player without knowledge.

Keywords

  • Gaussian Process
  • Voronoi Diagram
  • Gravity Center
  • Balance Cell
  • Bandit Problem

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Ahn, H.-K., Cheng, S.-W., Cheong, O., Golin, M., van Oostrum, R.: Competitive facility location: the Voronoi game. Theoretical Computer Science 310(1-3), 457–467 (2004)

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Anuth, J.: Strategien fur das Voronoi-spiel. Master’s thesis, FernUniveristät in Hagen (July 2007)

    Google Scholar 

  3. Auer, P., Ortner, R., Szepesvári, C.: Improved Rates for the Stochastic Continuum-Armed Bandit Problem. In: Bshouty, N., Gentile, C. (eds.) COLT 2007. LNCS (LNAI), vol. 4539, pp. 454–468. Springer, Heidelberg (2007)

    CrossRef  Google Scholar 

  4. Aurenhammer, F.: Voronoi diagrams: a survey of a fundamental geometric data structure. ACM Computing Surveys 23(3), 345–405 (1991)

    CrossRef  Google Scholar 

  5. Bouzy, B.: Associating domain-dependent knowledge and Monte-Carlo approaches within a go playing program. Information Sciences 175(4), 247–257 (2005)

    CrossRef  Google Scholar 

  6. Brochu, E., Cora, V., de Freitas, N.: A tutorial on bayesian optimization of expensive cost functions, with application to active user modeling and hierarchical reinforcement learning. Technical Report 23, Univ. of Brit. Col. (2009)

    Google Scholar 

  7. Bubeck, S., Munos, R., Stoltz, G., Szepesvári, C.: X-armed bandits. Journal of Machine Learning Research 12, 1655–1695 (2011)

    Google Scholar 

  8. Chaslot, G.: Monte-Carlo Tree Search. PhD thesis, Maastricht Univ (2010)

    Google Scholar 

  9. Chaslot, G., Winands, M., van den Herik, J., Uiterwijk, J., Bouzy, B.: Progressive strategies for Monte-Carlo tree search. New Mathematics and Natural Computation 4(3), 343–357 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Cheong, O., Har-Peled, S., Linial, N., Matoušek, J.: The one-round Voronoi game. In: 18th Symposium on Computational Geometry, pp. 97–101. ACM (2002)

    Google Scholar 

  11. Couëtoux, A., Hoock, J.-B., Sokolovska, N., Teytaud, O., Bonnard, N.: Continuous Upper Confidence Trees. In: Coello, C.A.C. (ed.) LION 2011. LNCS, vol. 6683, pp. 433–445. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  12. Coulom, R.: Efficient Selectivity and Backup Operators in Monte-Carlo Tree Search. In: van den Herik, H.J., Ciancarini, P., Donkers, H.H.L.M(J.) (eds.) CG 2006. LNCS, vol. 4630, pp. 72–83. Springer, Heidelberg (2007)

    CrossRef  Google Scholar 

  13. Dehne, F., Klein, R., Seidel, R.: Maximizing a Voronoi Region: The Convex Case. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 624–634. Springer, Heidelberg (2002)

    CrossRef  Google Scholar 

  14. Faidley, M., Poultney, C., Shasha, D.: The Voronoi game, http://home.dti.net/crispy/Voronoi.html

  15. Fekete, S.P., Meijer, H.: The one-round Voronoi game replayed. Computational Geometry Theory Appl. 30, 81–94 (2005)

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. Fortune, S.: A sweepline algorithm for Voronoi diagrams. Algorithmica 2(2), 153–174 (1987)

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. Gelly, S., Silver, D.: Achieving master level play in 9x9 computer go. In: AAAI, pp. 1537–1540 (2008)

    Google Scholar 

  18. Guibas, L.J., Stolfi, J.: Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams. In: 15th ACM Symposium on Theory Of Computing, pp. 221–234. ACM (1983)

    Google Scholar 

  19. Kleinberg, R.: Nearly tight bounds for the continuum-armed bandit problem. In: NIPS 17, pp. 697–704. MIT Press (2005)

    Google Scholar 

  20. Kocsis, L., Szepesvári, C.: Bandit Based Monte-Carlo Planning. In: Fürnkranz, J., Scheffer, T., Spiliopoulou, M. (eds.) ECML 2006. LNCS (LNAI), vol. 4212, pp. 282–293. Springer, Heidelberg (2006)

    CrossRef  Google Scholar 

  21. Lloyd, S.P.: Least squares quantization in PCM. IEEE Transactions on Information Theory 28, 129–137 (1982)

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. Mostafavi, M.A., Gold, C., Dakowicz, M.: Delete and insert operations in Voronoi/Delaunay methods and applications. Computers and Geosciences 29(4), 523–530 (2003)

    CrossRef  Google Scholar 

  23. Edward Rasmussen, C., Williams, C.K.I.: GaussianProcesses for Machine Learning. MIT Press (2006)

    Google Scholar 

  24. Selimi, I.: The Voronoi game (2008), http://www.voronoigame.com/

  25. Shewchuk, J.: Triangle: Engineering a 2d Quality Mesh Generator and Delaunay Triangulator. In: Lin, M.C., Manocha, D. (eds.) FCRC-WS 1996 and WACG 1996. LNCS, vol. 1148, pp. 203–222. Springer, Heidelberg (1996)

    CrossRef  Google Scholar 

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Bouzy, B., Métivier, M., Pellier, D. (2012). MCTS Experiments on the Voronoi Game. In: van den Herik, H.J., Plaat, A. (eds) Advances in Computer Games. ACG 2011. Lecture Notes in Computer Science, vol 7168. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31866-5_9

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  • DOI: https://doi.org/10.1007/978-3-642-31866-5_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31865-8

  • Online ISBN: 978-3-642-31866-5

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