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Electronic Colloquium on Computational Complexity

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Reports tagged with parity decision tree:
TR16-044 | 21st March 2016
Kaave Hosseini, Shachar Lovett

Structure of protocols for XOR functions

Revisions: 1

Let $f:\{0,1\}^n \to \{0,1\}$ be a boolean function. Its associated XOR function is the two-party function $f_\oplus(x,y) = f(x \oplus y)$.
We show that, up to polynomial factors, the deterministic communication complexity of $f_{\oplus}$ is equal to the parity decision tree complexity of $f$.
This relies on a novel technique ... more >>>

TR18-174 | 19th October 2018
Anastasiya Chistopolskaya, Vladimir Podolskii

Parity Decision Tree Complexity is Greater Than Granularity

Revisions: 2

We prove a new lower bound on the parity decision tree complexity $D_{\oplus}(f)$ of a Boolean function $f$. Namely, granularity of the Boolean function $f$ is the smallest $k$ such that all Fourier coefficients of $f$ are integer multiples of $1/2^k$. We show that $D_{\oplus}(f)\geq k+1$.

This lower bound is ... more >>>

TR21-046 | 22nd March 2021
Uma Girish, Avishay Tal, Kewen Wu

Fourier Growth of Parity Decision Trees

We prove that for every parity decision tree of depth $d$ on $n$ variables, the sum of absolute values of Fourier coefficients at level $\ell$ is at most $d^{\ell/2} \cdot O(\ell \cdot \log(n))^\ell$.
Our result is nearly tight for small values of $\ell$ and extends a previous Fourier bound ... more >>>

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