Herein, we investigate the randomized complexity, which is the least cost against the worst input, of AND-OR tree computation by imposing various restrictions on the algorithm to find the Boolean value of the root of that tree and no restrictions on the tree shape. When a tree satisfies a certain condition regarding its symmetry, directional algorithms proposed by Saks and Wigderson (1986), special randomized algorithms, are known to achieve the randomized complexity. Furthermore, there is a known example of a tree that is so unbalanced that no directional algorithm achieves the randomized complexity (Vereshchagin 1998). In this study, we aim to identify where deviations arise between the general randomized Boolean decision tree and its special case, directional algorithms. In this paper, we show that for any AND-OR tree, randomized depth-first algorithms, which form a broader class compared with directional algorithms, have the same equilibrium as that of the directional algorithms. Thus, we get the collapse result on equilibria inequalities that holds for an arbitrary AND-OR tree. This implies that there exists a case where even depth-first algorithms cannot be the fastest, leading to the separation result on equilibria inequality. Additionally, a new algorithm is introduced as a key concept for proof of the separation result.

翻译：本文研究AND-OR树计算的随机化复杂度（即针对最坏输入的最小成本），其中对求解树根布尔值的算法施加多种限制，但对树形不作任何约束。当树满足特定对称性条件时，Saks与Wigderson（1986）提出的定向算法（一类特殊随机化算法）已知可实现随机化复杂度。此外，存在已知案例表明：当树极度不平衡时，任何定向算法均无法达到随机化复杂度（Vereshchagin 1998）。本研究旨在揭示一般随机化布尔决策树与其特例（定向算法）产生偏差的根源。本文证明：对于任意AND-OR树，较定向算法更广义的随机化深度优先算法具有与定向算法相同的均衡性，由此得到适用于任意AND-OR树的均衡不等式坍缩结果。这暗示存在深度优先算法亦非最优解的情形，进而推导出均衡不等式的分离结果。此外，本文引入一种新算法作为证明分离结果的关键概念。