Separation and Collapse of Equilibria Inequalities on AND-OR Trees without Shape Constraints
Fuki Ito, Toshio Suzuki
TL;DR
This work proves that for any AND-OR tree, the randomized depth-first complexity $R_{\mathrm{DF}}(T)$ matches the directional-based bound $d(T)$, thereby collapsing a longstanding hierarchy of equilibria and showing that depth-first strategies can be as powerful as specialized directional approaches. The authors introduce a novel chimera construction $B_{\alpha,\delta}$ that blends a depth-first base algorithm with directional components and a mixed distribution $S^{\delta}$ to bridge DF and dir classes. They establish key cost-inequality tools and prove, by combining two problems, that $R_{\mathrm{DF},i}(T)=R_{\mathrm{dir},i}(T)=d_i(T)$ for both root values $i\in\{0,1\}$, and hence $R_{\mathrm{DF}}(T)=R_{\mathrm{dir}}(T)=d(T)$. A striking corollary is the existence of trees where $R(T)<R_{\mathrm{DF}}(T)$, highlighting that depth-first algorithms may fail to achieve the optimal zero-error randomized complexity in some cases. The results deepen understanding of randomized decision trees and have implications for when depth-first evaluation can be optimal or suboptimal in AND-OR tree computation.
Abstract
Herein, we investigate the zero-error 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. 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.