An analysis of the conspiracy numbers algorithm

Abstract

McAllester's conspiracy numbers algorithm is a minimax search procedure that builds game trees to variable depths without application-dependent knowledge. The algorithm gathers information to determine how likely it is that the search of a sub-tree will produce a useful result. “Likeliness” is measured by the conspiracy numbers, the minimum number of leaf nodes that must change their value (by being searched deeper) to cause the minimax value of a sub-tree to change. The search is controlled by the conspiracy threshold ( CT), the minimum number of conspirators beyond which it is considered unlikely that a sub-tree's value can be changed. This paper analyzes the best case performance for the algorithm. McAllester's original algorithm is shown to build a tree of size O( w ( CT)/( w − 1) ), where w is the branching factor of the tree. A new improvement to the algorithm is shown to reduce this complexity to O( CT 2). Hence, for a given fixed w, the algorithm's best case can be improved from exponential to quadratic growth. Although the minimal tree case is not necessarily representative of the search trees built in practice, experimental results are presented that demonstrate the improvement appears to be significant in the expected case as well.