MTD-f

MTD(f) is a minimax search algorithm developed in 1994 by Aske Plaat, Jonathan Schaeffer, Wim Pijls, and Arie de Bruin. Experiments with tournament-quality chess, checkers, and Othello programs show it to be a highly efficient minimax algorithm. The name MTD(f) is an abbreviation for MTD(n,f) (Memory-enhanced Test Driver with node n and value f). It is an alternative to the alpha-beta pruning algorithm.

Origin

MTD(f) was first described in a University of Alberta Technical Report authored by Aske Plaat, Jonathan Schaeffer, Wim Pijls, and Arie de Bruin, which would later receive the ICCA Novag Best Computer Chess Publication award for 1994/1995. The algorithm MTD(f) was created out of a research effort to understand the SSS* algorithm, a best-first search algorithm invented by George Stockman in 1979. SSS* was found to be equivalent to a series of alpha-beta calls, provided that alpha-beta used storage, such as a well-functioning transposition table.

The name MTD(f) stands for Memory-enhanced Test Driver, referencing Judea Pearl's Test algorithm, which performs Zero-Window Searches. MTD(f) is described in depth in Aske Plaat's 1996 PhD thesis.

Zero-window searches

MTD(f) derives its efficiency by only performing zero-window alpha-beta searches, with a "good" bound (variable beta). In NegaScout, the search is called with a wide search window, as in AlphaBeta(root, −INFINITY, +INFINITY, depth), so the return value lies between the value of alpha and beta in one call. In MTD(f), AlphaBeta fails high or low, returning a lower bound or an upper bound on the minimax value, respectively. Zero-window calls cause more cutoffs, but return less information - only a bound on the minimax value. To find the minimax value, MTD(f) calls AlphaBeta a number of times, converging towards it and eventually finding the exact value. A transposition table stores and retrieves the previously searched portions of the tree in memory to reduce the overhead of re-exploring parts of the search tree.

Pseudocode

function MTDF(root, f, d) is
    g := f
    upperBound := +∞
    lowerBound := −∞

    while lowerBound < upperBound do
        β := max(g, lowerBound + 1)
        g := AlphaBetaWithMemory(root, β − 1, β, d)
        if g < β then
            upperBound := g 
        else
            lowerBound := g

    return g

f

First guess for best value. The better the quicker the algorithm converges. Could be 0 for first call.

d

Depth to loop for. An iterative deepening depth-first search could be done by calling MTDF() multiple times with incrementing d and providing the best previous result in f.

AlphaBetaWithMemory is a variation of Alpha Beta Search that caches previous results.

Performance

NegaScout calls the zero-window searches recursively. MTD(f) calls the zero-window searches from the root of the tree. Implementations of the MTD(f) algorithm have been shown to be more efficient (search fewer nodes) in practice than other search algorithms (e.g. NegaScout) in games such as chess [1], checkers, and Othello. For search algorithms such as NegaScout or MTD(f) to perform efficiently, the transposition table must work well. Otherwise, for example when a hash-collision occurs, a subtree will be re-expanded. When MTD(f) is used in programs suffering from a pronounced odd-even effect, where the score at the root is higher for even search depths and lower for odd search depths, it is advisable to use separate values for f to start the search as close as possible to the minimax value. Otherwise, the search would take more iterations to converge on the minimax value, especially for fine grained evaluation functions.

Zero-window searches hit a cut-off sooner than wide-window searches. They are therefore more efficient, but, in some sense, also less forgiving, than wide-window searches. Because MTD(f) only uses zero-window searches, while Alpha-Beta and NegaScout also use wide window searches, MTD(f) is more efficient. However, wider search windows are more forgiving for engines with large odd/even swings and fine-grained evaluation functions. For this reason some chess engines have not switched to MTD(f). In tests with tournament-quality programs such as Chinook (checkers), Phoenix (chess), and Keyano (Othello), the MTD(f) algorithm outperformed all other search algorithms.

Recent algorithms like Best Node Search is suggested to outperform MTD(f).

External links

• Description of MTD(f) algorithm
• Description of the Cilkchess massively parallel chess program at MIT