A. Plaat
http://repub.eur.nl/ppl/7450/
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RePub, Erasmus University RepositoryA theory of game trees, based on solution trees
http://repub.eur.nl/pub/468/
Mon, 01 Jan 1996 00:00:01 GMT<div>W.H.L.M. Pijls</div><div>A. de Bruin</div><div>A. Plaat</div>
In this paper a complete theory of game tree algorithms is presented, entirely based upon the notion of a solution tree. Two types of solution trees are distinguished: max and min solution trees respectively. We show that most game tree algorithms construct a superposition of a max and a min solution tree. Moreover, we formulate a general cut-off criterion in terms of solution trees. In the second half of this paper four well known algorithms, viz., alphabeta, SSS*, MTD and Scout are studied extensively. We show how solution trees feature in these algorithms and how the cut-off criterion is applied.A new paradigm for minimax search
http://repub.eur.nl/pub/1440/
Sun, 01 Jan 1995 00:00:01 GMT<div>A. Plaat</div><div>J. Schaeffer</div><div>W.H.L.M. Pijls</div><div>A. de Bruin</div>
This paper introduces a new paradigm for minimax game-tree search algorithms. MT is a memory-enhanced version of Pearl's Test procedure. By changing the way MT is called, a number of best-first game-tree search algorithms can be simply and elegantly constructed (including SSS*).
Most of the assessments of minimax search algorithms have been based on simulations.
However, these simulations generally do not address two of the key ingredients of high
performance game-playing programs: iterative deepening and memory usage. This paper
presents experimental data from three game-playing programs (checkers, Othello and chess),
covering the range from low to high branching factor. The improved move ordering due to
iterative deepening and memory usage results in significantly different results from those
portrayed in the literature. Whereas some simulations show alpha-beta expanding almost
100% more leaf nodes than other algorithms [Marsland, Reinefeld & Schaeffer, 1987],
our results showed variations of less than 20%.
One new instance of our framework MTD(f) out-performs our best alpha-beta searcher
(aspiration NegaScout) on leaf nodes, total nodes and execution time. To our knowledge,
these are the first reported results that compare both depth-first and best-first algorithms given the same amount of memory.SSS* = AB+TT
http://repub.eur.nl/pub/1441/
Sun, 01 Jan 1995 00:00:01 GMT<div>A. Plaat</div><div>J. Schaeffer</div><div>W.H.L.M. Pijls</div><div>A. de Bruin</div>
In 1979 Stockman introduced the SSS* minimax search algorithm that dominates alpha-beta
in the number of leaf nodes expanded. Further investigation of the algorithm showed that it had three serious drawbacks, which prevented its use by practitioners: it is difficult to understand, it has large memory requirements, and it is slow. This paper presents an alternate formulation of SSS*, in which it is implemented as a series of alpha-beta calls that use a transposition table (ABSSS). The reformulation solves all three perceived drawbacks of SSS*, making it a practical algorithm. Further, because the search is now based on alpha-beta, the extensive research on minimax search enhancements can be easily integrated into ABSSS.
To test ABSSS in practise, it has been implemented in three state-of-the-art programs: for checkers, Othello and chess. ABSSS is comparable in performance to alpha-beta on leaf node count in all three games, making it a viable alternative to alpha-beta in practise.
Whereas SSS* has usually been regarded as being entirely different from alpha-beta, it
turns out to be just an alpha-beta enhancement, like null-window searching. This runs
counter to published simulation results. Our research leads to the surprising result that
iterative deepening versions of alpha-beta can expand fewer leaf nodes than iterative
deepening versions of SSS* due to dynamic move re-ordering.Solution trees as a basis for game tree search
http://repub.eur.nl/pub/1456/
Sat, 01 Jan 1994 00:00:01 GMT<div>A. de Bruin</div><div>W.H.L.M. Pijls</div><div>A. Plaat</div>
A game tree algorithm is an algorithm computing the minimax value of the root of a game tree. Many algorithms use the notion of establishing proofs that this value lies above or below some boundary value. We show that this amounts to the construction of a solution tree. We discuss the role of solution trees and critical trees in the following algorithms: Principal Variation Search, alpha-beta, and SSS-2. A general procedure for the
construction of a solution tree, based on alpha-beta and Null-Window-Search, is given.
Furthermore two new examples of solution tree-based algorithms are presented, that surpass
alpha-beta, i.e., never visit more nodes than alpha-beta, and often less.