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A transposition table of sufficientsize does the job. Storing nodes in memory gets rid of the overhead inherent inmultiple re-searches.The boundsdelimit the range of possible values for the minimax value.Each time MTD( f) calls AlphaBeta it gets a value back thatnarrows the range, and the algorithm is one step closer to hitting the minimaxvalue. At the root of the treethe return bounds are stored in upperbound (after AlphaBeta'failed low') and lowerbound (after AlphaBeta 'failed high'). Zero window AlphaBeta calls return bounds.

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The narrower the AlphaBeta window, the more cutoffs you get,the more efficient the search is.Hence MTD( f) uses only search windows of zero size.

To summarize, the core ideas of MTD( f) are: Sometimescoarse grain functions work better than fine grain, both for NegaScoutand MTD( f). Also, it can be quite instructive toexperiment with different evaluation function grain sizes. (Don Dailey found that a scheme like this works well ina version of Cilkchess.) At the end, if you overshoot the minimaxvalue, you have to make a small search in the opposite direction,using the previous search bound without an extra bonus, tomake the final convergence. It may help to dynamicallyincrease the step size: instead of using the previous bound, one can,for example, add an extra few points in the search direction (for failinghigh, or searching upward, adding the bonus, and for failing low,or searching downward, subtracting the bonus) every two passes or so. Big score swings canbecome inefficient in for these programs. Some programs have afine grained evaluation function, where positional knowledge can beworth as little as one hundredst of a pawn. The coarser the grain of eval, the less passes MTD( f)has to make to converge to the minimax value.

In the interest ofbrevity that is not shown in this code. In a realprogram, you would also store the best move in the transpositiontable, and upon retrieving search it first. The store function is needed to make sure thatthe table is filled with values as they become available. The lines around retrieve makesure that if a value is present in the table, it is used, instead ofcontinuing the search. Return g Transposition table access takes place in the retrieve and store calls. If g alpha and g= beta then n.lowerbound:= g store n.lowerbound * Fail low result implies an upper bound */ * Traditional transposition table storing of bounds */ G := min( g, AlphaBetaWithMemory( c, alpha, b, d- 1)) If n.upperbound alpha) and ( c != NOCHILD) do If n.lowerbound >= beta then return n.lowerbound Ifretrieve( n) OK then /* Transposition table lookup */ function AlphaBetaWithMemory( n : node_type alpha, beta, d : integer) : integer

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The transposition table access code is the sameas what is used in most tournament chess, checkers, and Othello programs.

To be sure, here's a minimax version of the pseudo code ofAlphaBetaWithMemory. An ordinary tranposition table ofreasonable size suffices, as our experiments showed (see furtherreading). In order for MTD( f) to be efficient yourAlphaBeta has to store the nodes it has searched. If AlphaBetawouldn't do that, then each pass of MTD( f) would re-exploremost of those nodes. AlphaBetaWithMemoryNote that the MTD( f) code calls an AlphaBeta version that stores its nodes in memory as it has determined their value, andretrieving their values in a re-search. But still, Weill's work indicatesthat it is worthwhile to experiment with variants on MTD( f)'schoice of pivot value leaving ample room for more research. On the down side, bisection yields a valuefor the search window, beta, that turns out to be not asefficient as MTD( f)'s choice. In MTD terms the idea of C* is to bisect theinterval formed by the upper and lower bounds, reducing the number ofAlphaBetaWithMemory calls. Jean-Christophe Weill has published a number of papers on experiments with anegamax version of C*. Another instance of the MTD framework is equivalent to theK.