# 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} }

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

#### Figures and Topics from this paper

#### 114 Citations

A Minimax Algorithm Better Than Alpha-Beta? Yes and No

- Mathematics, Computer Science
- Artif. Intell.
- 1983

Numerical comparison of the expected complexities of the two algorithms is finally carried out over a wide spectrum of search depths and branching degrees and shows that the savings in the number of positions evaluated by SSS^* relative to that of @a-@b is rather limited and is not enough to offset the increase in other computational resources. Expand

A Minimax Algorithm Better Than Alpha-beta?: No and Yes

- Computer Science
- ArXiv
- 2017

A new formulation for Stockman's SSS* algorithm, based on Alpha-Beta, is presented, solving all the perceived drawbacks of SSS*, finally transforming it into a practical algorithm. Expand

A Minimax Algorithm Better than Alpha-Beta ?

- 2010

This paper has three main contributions to our understanding of fixed-depth minimax search: (A) A new formulation for Stockman’s SSS* algorithm, based on Alpha-Beta, is presented. It solves all the… Expand

Generalization of Alpha-Beta and SSS Search Procedures

- Mathematics, Computer Science
- Artif. Intell.
- 1986

A general framework is provided, within which various conventional procedures including alpha-beta and SSS∗ can be naturally generalized to the informed model, which permits the usage of heuristic information pertaining to nonterminal nodes. Expand

Equilibrium points of an AND-OR tree: Under constraints on probability

- Mathematics, Computer Science
- Ann. Pure Appl. Log.
- 2015

A depth-first search based algorithm to find the truth value of the root of an AND–OR tree is considered, and it is shown that the maximizer is an independent identical distributions (IID). Expand

Independent Distributions on a Multi-Branching AND-OR Tree of Height 2

- Mathematics, Computer Science
- Discret. Appl. Math.
- 2020

This paper asks whether Tarsi-type result holds for an AND-OR tree of height 2, and investigates the case where d is an independent distribution and probability depends on each leaf, and performs induction on the number of leaves. Expand

Non-Depth-First Search against Independent Distributions on an AND-OR Tree

- Mathematics, Computer Science
- Inf. Process. Lett.
- 2018

A key to the proof is that if ID d achieves the equilibrium among IDs then d has an optimal algorithm that is depth-first, which is extended to multi-branching trees to the case where non-depth-first algorithms are taken into consideration. Expand

A randomized competitive algorithm for evaluating priced AND/OR trees

- Computer Science, Mathematics
- Theor. Comput. Sci.
- 2008

A 56@m(T)-competitive randomized polynomial time algorithm for an AND/OR tree T, which contrasts with the best known lower bound @m (T)/2. Expand

A pr 2 01 8 Independent Distributions on a Multi-Branching ANDOR Tree of Height 2

- 2018

We investigate an AND-OR tree T and a probability distribution d on the truth assignments to the leaves. Tarsi (1983) showed that if d is an independent and identical distribution (IID) such that… Expand

The Eigen Distribution of an AND-OR Tree under Directional Algorithms

- Mathematics
- 2012

Consider a probability distribution d on the truth assignments to a perfect binary AND-OR tree. Liu and Tanaka (2007) extends the work of Saks and Wigderson (1986), and they characterize the… Expand

#### References

SHOWING 1-10 OF 17 REFERENCES

On the Branching Factor of the Alpha-Beta Pruning Algorithm

- Mathematics, Computer Science
- Artif. Intell.
- 1978

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

- Mathematics
- 1973

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 bottom… Expand

A Minimax Algorithm Better than Alpha-Beta?

- Computer Science
- Artif. Intell.
- 1979

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

- Mathematics, Computer Science
- Artif. Intell.
- 1980

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

- Mathematics, Computer Science
- J. Algorithms
- 1981

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

- Computer Science
- Artif. Intell.
- 1975

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

- Computer Science
- JACM
- 1969

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

- Computer Science
- JACM
- 1983

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 that… Expand

Permutation enumeration: four new permutation algorithms

- Mathematics, Computer Science
- CACM
- 1976

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

- Mathematics, Computer Science
- JACM
- 1973

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 above… Expand