Negascout
NegaScout or Principal Variation Search is a negamax algorithm that can be faster than alpha-beta pruning. Like alpha-beta pruning, NegaScout is a directional search algorithm for computing the minimax value of a node in a tree. It dominates alpha-beta pruning in the sense that it will never examine a node that can be pruned by alpha-beta; however it relies on accurate move ordering to capitalize on this advantage.
NegaScout works best when there is a good move ordering. In practice, the move ordering is often determined by previous shallower searches. It produces more cutoffs than alpha-beta by assuming that the first explored node is the best. In other words, it supposes the first node is in the principal variation. Then, it can check whether that is true by searching the remaining nodes with a null window (also known as a scout window; when alpha and beta are equal), which is faster than searching with the regular alpha-beta window. If the proof fails, then the first node was not in the principal variation, and the search continues as normal alpha-beta. Hence, NegaScout works best when the move ordering is good. With a random move ordering, NegaScout will take more time than regular alpha-beta; although it will not explore any nodes alpha-beta did not, it will have to re-search many nodes.
In chess engines, NegaScout has typically given a 10 percent performance increase[citation needed].
Alexander Reinefeld invented NegaScout several decades after the invention of alpha-beta pruning. He gives a proof of correctness of NegaScout in his book.[1]
Another search algorithm called MTD(f) can theoretically result in even fewer nodes searched. However it has practical issues (in particular, it relies heavily on the transposition table) and nowadays most chess engines still use a form of NegaScout in their search. Yet another search algorithm which tends to do better than NegaScout in practice is the best-first algorithm called SSS*, although neither algorithm dominates the other. There are trees in which NegaScout searches fewer nodes than SSS* and vice-versa. However, note that SSS* is not a depth-first search and thus has larger memory requirements.
Negascout takes after SCOUT, invented by Judea Pearl in 1980, which was the first algorithm to outperform alpha-beta and to be proven asymptotically optimal[2][3]. Null windows, with β=α+1 in a negamax setting, were invented independently by J.P. Fishburn and used in an algorithm similar to SCOUT in an appendix to his Ph.D. thesis [4], in a parallel alpha-beta algorithm[5], and on the last subtree of a search tree root node[6]
[edit] Pseudocode
function negascout(node, depth, α, β)
if node is a terminal node or depth = 0
return the heuristic value of node
b := β (* initial window is (-β, -α) *)
foreach child of node
score := -negascout (child, depth - 1, -b, -α)
if α < score < β and child is not first child (* check if null-window failed high *)
score := -negascout(child, depth - 1, -β, -α) (* full re-search *)
α := max(α, score)
if α ≥ β
return α (* Beta cut-off *)
b := α + 1 (* set new null window *)
return α
[edit] References
- ^ A. Reinefeld. Spielbaum-Suchverfahren. Informatik-Fachbericht 200, Springer-Verlag, Berlin (1989), ISBN 3-540-50742-6
- ^ Pearl, J., "SCOUT: A Simple Game-Searching Algorithm With Proven Optimal Properties," Proceedings of the First Annual National Conference on Artificial Intelligence, Stanford University, August 18-21, 1980, pp. 143-145.
- ^ Pearl, J., "Asymptotic Properties of Minimax Trees and Game-Searching Procedures," Artificial Intelligence, Vol. 14, No. 2, pp. 113-138, September 1980.
- ^ Fishburn, J.P., "Analysis of Speedup in Distributed Algorithms", UMI Research Press ISBN 0-8357-1527-2, 1981, 1984.
- ^ Fishburn, J.P., Finkel, R.A., and Lawless, S.A., "Parallel Alpha-Beta Search on Arachne" Proceedings 1980 Int. Conf. Parallel Processing, IEEE, August 26-29, 1980, pp. 235-243.
- ^ Fishburn, J.P., "An Optimization of Alpha-Beta Search" ACM SIGART Bulletin, issue 72, July 1980, pp 29-31.
