Trends in game tree search

  • Arie de Bruin
  • Wim Pijls
Invited Papers Programming and Software Engineering
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1175)


This paper deals with algorithms searching trees generated by two-person, zero-sum games with perfect information. The standard algorithm in this field is Alpha-Beta. We will discuss this algorithm as well as extensions, like transposition tables, iterative deepening and NegaScout. Special attention is devoted to domain knowledge pertaining to game trees, more specifically to solution trees.

The above mentioned algorithms implement depth first search. The alternative is best first search. The best known algorithm in this area is Stock-man's SSS*. We treat a variant equivalent to SSS* called SSS-2. These algorithms are provably better than Alpha-Beta, but it needs a lot of tweaking to show this in practice. A variant of SSS-2, cast in Alpha-Beta terms, will be discussed which does realize this potential. This algorithm is however still worse than NegaScout. On the other hand, applying a similar idea as the one behind NegaScout to this last SSS version yields the best (sequential) game tree searcher known up till now: MTD(f).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. M. Baudet, On the branching factor of the alpha-beta pruning algorithm. Artificial Intelligence 10 (1978), pp 173–199.Google Scholar
  2. 2.
    Subir Bhattacharya and A. Bagchi, A faster alternative to SSS * with extension to variable memory, Information processing letters 47 (1993), 209–214.Google Scholar
  3. 3.
    Toshihide Ibaraki, Generalization of alpha-beta and SSS * search procedures, Artificial Intelligence 29 (1986), 73–117.Google Scholar
  4. 4.
    V. Kumar and L.N. Kanal, A General Branch and Bound Formulation for Understanding and Synthesizing And/Or Tree Search Procedures, Artificial Intelligence 21 (1983), 179–198.Google Scholar
  5. 5.
    Donald E. Knuth and Ronald W. Moore, An analysis of alpha-beta pruning, Artificial Intelligence 6 (1975), no. 4, 293–326.Google Scholar
  6. 6.
    T. A. Marsland, A. Reineveld, J. Schaeffer, Low Overhead Alternatives to SSS *, Artificial Intelligence 31 (1987) pp. 185–199.Google Scholar
  7. 7.
    Wim Pijls and Arie de Bruin, Another view on the SSS * algorithm, Algorithms, International Symposium SIGAL '90, Tokyo, Japan, August 16–18, 1990 Proceedings (T. Asano, T. Ibaraki, H. Imai, and T. Nishizeki, eds.), LNCS, vol. 450, Springer-Verlag, August 1990, pp. 211–220.Google Scholar
  8. 8.
    Wim Pijls. Shortest Paths and Game Trees. PhD Thesis, Erasmus University Rotterdam, The Netherlands, November 1991.Google Scholar
  9. 9.
    Wim Pijls and Arie de Bruin, Searching informed game trees, In: Algorithms and Computation, ISAAC 92 (T. Ibaraki, ed), pp. 332–341, LNCS 650.Google Scholar
  10. 10.
    Wim Pijls and Arie de Bruin, A theory of game trees based on solution trees, Tech.Rep. EUR-CS-96-xxx, Erasmus University Rotterdam, 1996.Google Scholar
  11. 11.
    Aske Plaat. Research Re:Search & Research. PhD Thesis, Erasmus University Rotterdam, The Netherlands, June 1996.Google Scholar
  12. 12.
    Aske Plaat, Jonathan Schaeffer, Wim Pijls and Arie de Bruin, A Minimax Algorithm Better than SSS *, In: Artificial Intelligence, to appear.Google Scholar
  13. 13.
    Aske Plaat, Jonathan Schaeffer, Wim Pijls and Arie de Bruin, Exploiting graph properties of game trees. In Proceedings of the 13th National Conference on Artificial Intelligence (AAAI '96), Portland, OR, August 1996. American Association for Artificial Intelligence, AAAI Press.Google Scholar
  14. 14.
    Alexander Reinefeld and Peter Ridinger. Time efficient state space search. Artificial Intelligence, 71 (2), pp. 397–408, 1994.Google Scholar
  15. 15.
    Igor Roizen and Judea Pearl, A minimax algorithm better than alpha-beta? yes and no, Artificial Intelligence 21 (1983), 199–230.Google Scholar
  16. 16.
    Jonathan Schaeffer, The history heuristic and alpha-beta search enhancements in practice, IEEE Transactions on Pattern Analysis and Machine Intelligence PAMI-11 (1989), no. 1, 1203–1212.Google Scholar
  17. 17.
    G. Stockman, A minimax algorithm better than alpha-beta?, Artificial Intelligence 12 (1979), no. 2, 179–196.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Arie de Bruin
    • 1
  • Wim Pijls
    • 1
  1. 1.Dept. Comp. ScienceErasmus UniversityThe Netherlands

Personalised recommendations