An Analysis of Alpha-Beta Priming'

The alpha-beta technique for searching game trees is analyzed, in an attempt to provide some insight into its behavior. The first portion o f this paper is an expository presentation o f the method together with a proof o f its correctness and a historical ch'scussion. The alpha-beta procedure is shown to be optimal in a certain sense, and bounds are obtained for its running time with various kinds o f random data. Put one pound of Alpha Beta Prunes in a jar or dish that has a cover. Pour one quart o f boiling water over prunes. The longer prunes soak, the plumper they get Alpha Beta Acme Markets, Inc., La Habra, California Computer programs for playing games like e, hess typically choos~ their moves by seaxching a large tree of potential continuations. A technique called "alpha-beta pruning" is generally ~ used to speed up such search processes without loss of information, The purpose of this paper is to analyze the alpha-beta procedure in order to obtain some quantitative estimates of its performance characteristics. i This research was supported in part by the National Science Foundation under grant number GJ 36473X and by the Office of Naval Research under contract NR 044-402. Artificial Intelligence 6 (1975), 293-326 Copyright © 1975 by North-Holland Publishing Company 294 D.E. KNUTH AND R. W. MOORE Section 1 defines the basic concepts associated with game trees. Section 2 presents the alpha-beta method together with a related technique which is similar, but not as powerful, because it fails to make "deep cutoffs". The correctness of both methods is demonstrated, and Section 3 gives examples and further development of the algorithms. Several suggestions for applying the method in practice appear in Section 4, and the history of alpha-beta pruning is discussed in Section 5. Section 6 begins the quantitative analysis, byderiving lower bounds on the amount of searching needed by alpha-beta and by any algorithm which solves the same general problem. Section 7 derives upper bounds, primarily by considering the case of random trees when no deep cutoffs are made. It is shown that the procedure is reasonably efficient even under these weak assumptions. Section 8 shows how to introduce some of the deep cutoffs into the analysis; and Section 9 shows that the efficiency improves when there are dependencies between successive moves. This paper is essentially selfcontained, except for a few mathematical resultsquoted i n the later sections. 1. Games and Position Values The two-person games we are dealing with can be characterized by a set of "positions", and by a set of rules for moving from one position to ~,nother, the players moving alternately. We assume that no infinite sequence of positions is allowed by the rules, 2 and that there are only finitely many legal moves from every position. It follows from the "infinity lemma" (see [11, Section 2.3,4.3]) that for every position p there is a number N(p) such that no game starting a t p lasts longer than N(p) moves. I fp is a position from which there are no legal moves, there is an integervalued function f(p) which represents the value of this position to the player whose turn it is to play from p; the value to the other player is assuraed to be f ( p ) . If p is a position from which there are d legal moves Pl, • •., Pd, where d > 1, the problem is to choose the "best" move. We assume that the best move is one which achieves the greatest possible value when the game ends, if the opponent also chooses moves which are best for h im. Let F(p) be the greatest possible value achievable from position p against the optimal defensive strategy, from the standpoint of the player who is moving from that 2 Strictly speaking, chess does not satisfy this condition, since its rules for repeated positions only give the players the option to request a draw; in certain circumstances; if neither player actually does ask for a draw,: the game can go on forever. But this technicality is of no practical importance, since computer chess programs only look finitely many moves ahead. I r i s possible to deal with infinite games by assigning appropriate values to repeated positions, but such questionsare beyond the scope of this paper. Artificial Intelligence 6 f1975), 293-326 AN ANALYSIS OF ALPHA-BETA PRUNING 295 position. Since the value.(to this player) after moving to position Pt will be -F(pl), we have ~f(p) if d = 0, (1) F(p) = (max(-F(pl),..., i f d > 0. This formula serves to define F(p) for al! positions p, by induction on the length of the longest game playable from p. In most discussions of game-playing, a slightly different formalism is used; the two players are named Max and Min, where all values are given from Max's viewpoint. Thus, if p is a terminal position with Max to move, its value is f(p) as before, but if p is a terminal position with Min to move its value is gO') = -f(P) . (2) Max will try to maximize the final value, and Min will try to minimize it. There are now two functions corresponding to( l) , namely F(p) V = ~f(P) if d = 0, [max(G(pl) , . . . , G(pd)) if d > 0, (3) which is the best value Max can guarantee starting at position p, and fg(p) if d = 0, G(p) = [.min(F(pl),..., F(Pd)) if d > 0, (4) which is the best that Min can be sure of achieving. As before, we assume that Pl , . •., Pa are the legal moves from position p. It is easy to prove by induction that the two definitions of F in (1) and (3) are identical, and that