## alpha-beta

Backtrack programming consists of searching a vector space of possibilities in a systematic fashion. For example, if actions a, b, c are to occur in that order and each action has two possibilities (say, {a1, a2}, {b1, b2}, {c1, c2}) then we might first consider performing the first of all possibilities, resulting in considering the sequence a1-b1-c1. We now iterate changing the last of the actions to some other possibility that has not yet been considered. This results in next considering a1-b1-c2. When we run out of possibilities of changing the last action, we change the last of the actions that has other possibilities. In this manner, we consider the sequences in the order: a1-b1-c1, a1-b1-c2, a1-b2-c1, a1-b2-c2, a2-b1-c1, a2-b1-c2, a2-b2-c1, a2-b2-c2. The set of sequences can be thought of as being the leaves of the tree of possibilities, where a tree node corresponds to a partial sequence, and the degree of a node is the number of possibilities of the next action.

Branch-and-bound is an embellishment of backtrack programming. Performing a sequence of actions results in reaching a leaf of the tree with which we can associate a value. Our desire is to determine whether there exists a sequence of actions which will lead to a leaf having at least (or at most) a specified value, goal, or to determine the maximum (or minimum) achievable value. During the process of the search, we may be able to assign upper and lower value bounds to intermediate nodes of the tree (such as after performing the partial sequence a1-b2) in such a manner that logically and mathematically it will always be the case that the value, v, of any leaf obtained by completing the partial sequence will have value between the upper and lower bounds. If the upper bound is less than goal or, for maximization problems, less than the highest value achieved thus far (or if the lower bound is more than goal or, for minimization problems, more than the lowest value achieved thus far) then we can avoid searching the collection of sequences that complete the partial sequence, thus pruning the search (and saving time).

Alpha-beta search is similar to branch-and-bound for a game between two opposing players or teams, in which one side has a goal of achieving or maximizing a value while the other side has a goal of preventing achievement or minimizing value.

There are a number of strategies that may improve the efficiency of alpha-beta search. Some of these amenities are effective for the bridge hand evaluation problem.

Dan Hirschberg
Computer Science Department
University of California, Irvine, CA 92697-3425
dan at ics.uci.edu