On the Decision Tree Complexity of Threshold Functions

Abstract

In this paper we study decision tree models with various types of queries. For a given function it is usually not hard to determine the complexity in the standard decision tree model (each query evaluates a variable). However in more general settings showing tight lower bounds is substantially harder. Threshold functions often have non-trivial complexity in such models and can be used to provide interesting examples.

Standard decision trees can be viewed as a computational model in which each query depends on only one input bit. In the first part of the paper we consider natural generalization of standard decision tree model: we address decision trees that are allowed to query any function depending on two input bits. We show the first lower bound of the form $n-o(n)$ for an explicit function (namely, the majority function) in this model. We also show that in the decision tree model with AND and OR queries of arbitrary fan-in the complexity of the majority function is $n-1$.

In the second part of the paper we address parity decision trees that are allowed to query arbitrary parities of input bits. There are various lower bound techniques for parity decision trees complexity including analytical techniques (degree over $\mathbb {F}_2$, Fourier sparsity, granularity) and combinatorial techniques (generalizations of block sensitivity and certificate complexity). These techniques give tight lower bounds for many natural functions. We give a new inductive argument tailored specifically for threshold functions. A combination of this argument with granularity lower bound allows us to provide a simple example of a function for which all previously known lower bounds are not tight.

The paper is partially supported by RFBR Grant 18-01-00822 and by HSE University Basic Research Program.