Upper and Lower Bounds on $T_1$ and $T_2$ Decision Tree Model

TL;DR

The paper analyzes generalized decision-tree models that allow subset queries, focusing on $T_1$ (lor) and $T_2$ variants. It establishes a strong lower bound for monotone graph properties in the $T_1$-model: detecting a non-tree requires at least $D_{\\lor}(g) \ge n \\log(n-2)$ queries, guided by Cayley’s formula and a union-of-trees argument. It also shows constructive upper bounds in the $T_2$-model: $MAJ_n$ can be computed in at most $\\frac{3n}{4}$ queries and $SYM_n$ in at most $n$ queries, using block-based strategies that combine $T_2$-queries within small variable blocks. Together, these results illuminate how expanding the set of allowable queries affects decision-tree complexity for both graph properties and symmetric boolean functions, with implications for related conjectures in query complexity under extended models.

Abstract

We study a decision tree model in which one is allowed to query subsets of variables. This model is a generalization of the standard decision tree model. For example, the $\lor-$decision (or $T_1$-decision) model has two queries, one is a bit-query and one is the $\lor$-query with arbitrary variables. We show that a monotone property graph, i.e. nontree graph is lower bounded by $n\log n$ in $T_1$-decision tree model. Also, in a different but stronger model, $T_2$-decision tree model, we show that the majority function and symmetric function can be queried in $\frac{3n}{4}$ and $n$, respectively.

Theorems & Definitions (8)

• Definition 1
• Definition 2
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof