The solution for the branching factor of the alpha-beta pruning algorithm and its optimality

@article{Pearl1982TheSF,
  title={The solution for the branching factor of the alpha-beta pruning algorithm and its optimality},
  author={Judea Pearl},
  journal={Commun. ACM},
  year={1982},
  volume={25},
  pages={559-564}
}
  • J. Pearl
  • Published 1982
  • Computer Science
  • Commun. ACM
M. Douglas Mcllroy* and Data Structures Editor stop(top). In such a situation, the element top has to be deleted from the stack and more operations are required to generate the next combination. When k > top > 2, one can show that the probability for a specific value of top that a[top] = stop(top) is a(top + l ) /a( top) , which reduces to (k top + l ) / (n top). Hence, when k is small compared to n, it is very unlikely that the next combination is generated by using the theoretical maxim u m… Expand
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References

SHOWING 1-10 OF 17 REFERENCES
On the Branching Factor of the Alpha-Beta Pruning Algorithm
  • G. Baudet
  • Mathematics, Computer Science
  • Artif. Intell.
  • 1978
TLDR
The branching factor of the alpha-beta pruning algorithm is shown to grow with n as @Q(n/lnn), therefore confirming a claim by Knuth and Moore that deep cut-offs only have a second order effect on the behavior of the algorithm. Expand
Analysis of the alpha-beta pruning algorithm
Abstract : Many game-playing programs must search very large game trees. Use of the alpha-beta pruning algorithm instead of the simple minimax search reduces by a large factor the number of bottomExpand
A Minimax Algorithm Better than Alpha-Beta?
Abstract An algorithm based on state space search is introduced for computing the minimax value of game trees. The new algorithm SSS∗ is shown to be more efficient than α-s in the sense that SSS∗Expand
Asymptotic Properties of Minimax Trees and Game-Searching Procedures
  • J. Pearl
  • Mathematics, Computer Science
  • Artif. Intell.
  • 1980
TLDR
It is shown that a game with WIN-LOSS terminals can be solved by examining, on the average, O [(d) h 2 ] terminal positions if positions if P 0 ≠ P∗ and O [(P∗ (1 − P ∗) ) h ] positionsif P 0 = P∷, the former performance being optimal for all search algorithms. Expand
A Space-Efficient On-Line Method of Computing Quantile Estimates
  • J. Pearl
  • Mathematics, Computer Science
  • J. Algorithms
  • 1981
TLDR
A recursive method of estimating ζ q based on the fact that if the terminal nodes of a uniform d -ary tree are assigned random values, independently drawn from a distribution F , then the minimax alue of the root node converges to a specified quantile of F for very tall trees is introduced. Expand
An Analysis of Alpha-Beta Pruning
TLDR
The alpha-beta procedure for searching game trees is shown to be optimal in a certain sense, and bounds are obtained for its running time with various kinds of random data. Expand
Experiments With Some Programs That Search Game Trees
TLDR
The problem of efficiently searching large trees is discussed, and a new method called “dynamic ordering” is described, and the older minimax and Alpha-Beta procedures are described for comparison purposes. Expand
Optimal Search on Some Game Trees
It is proved that the dlrecUonal algorithm for solving a game tree is optimal, in the sense of average run trine, for balanced trees (a family containing all uniform trees). This result implies thatExpand
Permutation enumeration: four new permutation algorithms
  • F. Ives
  • Mathematics, Computer Science
  • CACM
  • 1976
TLDR
Performance tests which have counted execution of assignment statements, comparisons, arithmetic operations, and subscripted array references have shown superiority of the new algorithms compared to Boothroyd's Implementation of M.B. Well's algorithm and Ehrlich's implementation of the Johnson-Trotter algorithm. Expand
Loopless Algorithms for Generating Permutations, Combinations, and Other Combinatorial Configurations
The purpose of this work is to find a method for building loopless algorithms for listing combinatorial items, like partitions, permutations, combinations. Gray code, etc. Algorithms for the aboveExpand
...
1
2
...