Information Acquisition in Minimal Window Search

The alpha-beta tree search algorithm can be improved through the use of minimal windows. Branches are searched with a minimal window [α,α+l] wi th the expectancy that this wi l l show the sub-tree to be inferior. If not, then that sub-tree must be re-searched. In this paper, several methods are discussed to minimize the cost of the re-search. Two new algorithms, INS and PNS, are introduced and their performance on practical trees is shown to be comparable to SSS*, but wi th considerably smaller overhead. 1 . I n t r o d u c t i o n The use of minimal windows [ l ] provides an improvement to the alpha-beta tree-searching algorithm (AB) [2], Minimal window search is based on the assumption that all subtrees are infe­ rior to the best subtree searched thus far, unt i l proven otherwise. Having searched the lirst branch with a full window [α,0], all remaining branches are searched with a minimal window [α,α+1], where a represents the best minimax value found so far. If the value returned is indeed < a, then our assumption was correct and the sub­ tree is inferior. Otherwise, this subtree is superior and usually must be re-searched wi th a wider win­ dow. The re-search idea originally appeared in Pearl's Scout algorithm [3]. Subsequently, there have been two generalizations, Principal Variat ion Search [4] and NegaSeout [5]. Figure 1 shows the NegaSeout (NS) algorithm for searching a tree of width w and depth d. If a node p is terminal, Evaluate(p) returns its value. For interior nodes, Gencrate(p) determines the w branches from p. Those branches whose minimal window search produces a better minimax value of v usually must be re-searched. Only when α < v < 0, and the remaining depth of search is greater than 2, is a re-search with a window [v,0] necessary. t Current address: Universitaet Hamburg, Farhbereich Informatik, Schlueterstr.70, D-2000 Hamburg 13, WestGermany. tt Research reported here was supported in part through Canadian NSERC grants A7902 and E5722. This paper introduces two new algorithms. Those use information acquired from the original search of a. subtree to minimize the cost of a possi­ ble re-search. I n f o r m e d NegaScou t ( INS) uses all available information to generate the smallest possible trees, but does so with increased storage overhead. P a r t i a l l y I n f o r m e d NegaScou t ( P N S ) is a compromise between NS and INS. The performance of NS, PNS, and INS is compared with AB and SSS* [6,7]. INS searches trees of size comparable to those traversed by SSS', but does so wi th lower overheads. A. Reinefeld et al. 1041 Information gathered from the ini t ial search of the subtree is used on a re-search to allow two new types of cut-offs. Figure 2 illustrates the ignore left cut-off. In Figure 2a, the subtree has been searched with a minimal-window of [100,101]. The descendant B returned a value of 105 causing a normal beta cut-off. If at some future point it is necessary to re-search this subtree, descendant A need not be looked at again since it has already been shown to be inferior to B, Figure 2b. Figure 3 illustrates the prove best cut-off. At these nodes, a beta cut-off has not occurred and all descendants have been examined. Each of the values returned is an upper bound on the subtree's true value, Figure 3a. If a re-search is necessary on this subtree, there are three things that can be done to minimize tree size. First of all, the branches can be re-ordered according to their values from the ini t ial search. By sorting the branches in descending order of value, the branch with the highest upper bound (and therefore with the highest probability of being the root of the best subtree) is searched first. Figure 3b. Secondly, since the ini t ial value for each sub­ tree represents an upper bound, the re-search can be done wi th a narrow window instead of a minimal-window. By doing this, no re-searches of re-searches can ever occur. Finally, if the search of a subtree returns a true value that is greater than the upper bound of any of the other descendants, then those descen­ dants can be discarded without any further work. For example, in Figure 3b, if move B is re-searched and returns a true value of 88, then moves A, and C need not be searched again, since their values can never exceed that of B. It turns out that ignore left cut-offs are just a special case of prove best cut-offs. Branches pro­ ven inferior can be treated as having value and the rest of the branches as having a +∞ value. Retrieving this information and performing a stable sort creates the prove best condit ion. The cut-offs are treated differently because in an actual implementation the ignore left cut-offs require less storage to maintain the necessary information, e.g. only the number i of the best descendant thus far need be saved. On a re-search, descendants 1 through i l are ignored and the remainder searched. At prove best nodes, the values for all descendants must be saved. 3 . A l g o r i t h m s NegaScout can be enhanced to use informa­ tion from the ini t ia l search of a subtree to aid in any re-searches. Every time a node is visited, a record is kept of the results obtained from search­ ing each descendant subtree. Either a beta cut-off occurs, and ignore left information is available for a re-search, or all descendants are examined, and prove best information is available. In both cases, this information can be linked together to form a map of the subtree just searched. If a re-search is necessary, the map data can be used to achieve ignore left and prove best cut-offs that are not pos­ sible in NegaScout. Informed NegaScout (INS), see Appendix, does exactly this for all nodes in a tree. The storage overhead in saving all this infor­ mation is proportional to w* entries, which is less than for SSS*. Nevertheless this may be too much, even if one reclaims storage whenever possi­ ble. As an alternative, Partially Informed NegaScout (PNS) has been implemented, provid­ ing a compromise between the complete informa­ tion of INS and the zero information of NS. One can devise many different compromise algorithms, our version of PNS only retains information about prove best cut-offs near the root of each subtree and maintains the principal variation, the path to the terminal node that the ini t ial search considered best. This algorithm tries to provide many of the benefits of INS without the storage overhead. An important point to note is that the infor­ mation used by INS is not a hash table or a tran­ sposition table [4). Whereas transposition tables are most useful in directed graphs, the methods described here are applicable to any tree structure and do not depend on the properties of the appli­ cation. 1042 A. Reinefeld al