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

TL;DR

This paper investigates bounds on decision tree models that allow querying subsets of variables, extending standard models, and establishes lower bounds and query complexities for specific functions.

Contribution

It introduces bounds on $T_1$ and $T_2$ decision tree models, generalizing standard decision trees and analyzing their query complexities for various functions.

Findings

Monotone property graphs have a lower bound of $n \,\log n$ in $T_1$-decision trees.

Majority functions can be queried in $\frac{3n}{4}$ steps in $T_2$-decision trees.

Symmetric functions can be queried in $n$ steps in $T_2$-decision trees.

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.