queries, and strengthens this bound from the worst-case over inputs to the average over inputs. The method also yields bounds for specific Boolean functions in terms of their spectra (their Fourier transforms). The simplest instance of this spectral bound yields the result (previously established by Feige, Peleg, Raghavan, and Upfal) that the parity function of n arguments requires

\Me(n \log n)

queries and again strengthens this bound from the worst-case over inputs to the average over inputs. In its full generality, the spectral bound applies to the "highly resilient" functions introduced by Chor, Friedman, Goldreich, Hastad, Rudich, and Smolensky, and it yields nonlinear lower bounds whenever the resiliency is asymptotic to the number of arguments.

DOI: 10.1137/s0097539796310102

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A Unified Lower Bound on the Noisy Query Complexity of Boolean Functions

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arXiv:2606.11448Jun 9, 20260 citations

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Yuzhou Gu, Xin Jun Li, Yinzhan Xu

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arXiv:2502.04632Feb 7, 20250 citations

Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture

Arkadev Chattopadhyay, Ankit Garg, Suhail Sherif

We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on $n$ input bits, each of which has approximate Fourier sparsity at most $O(n^3)$ and randomized parity decision tree complexity $Θ(n)$ . This improves upon the recent work of Chattopadhyay, Mande and Sherif (JACM '20) both qualitatively (in terms of designing a large number of examples) and quantitatively (improving the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on $n$ input bits, each of which has approximate Fourier sparsity at most $O(n^3)$ and randomized parity decision tree complexity $Θ(n)$. This improves upon the recent work of Chattopadhyay, Mande and Sherif (JACM '20) both qualitatively (in terms of designing a large number of examples) and quantitatively (improving the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).

arXiv:2009.02717Sep 6, 20201 citation

Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture

Arkadev Chattopadhyay, Ankit Garg, Suhail Sherif

We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on n input bits, each of which has approximate Fourier sparsity at most O(n³) and randomized parity decision tree complexity Θ(n). This improves upon the recent work of Chattopadhyay, Mande and Sherif [Chattopadhyay et al., 2020] both qualitatively (in terms of designing a large number of examples) and quantitatively (shrinking the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on n input bits, each of which has approximate Fourier sparsity at most O(n³) and randomized parity decision tree complexity Θ(n). This improves upon the recent work of Chattopadhyay, Mande and Sherif [Chattopadhyay et al., 2020] both qualitatively (in terms of designing a large number of examples) and quantitatively (shrinking the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).

Leibniz international proceedings in informaticsJan 1, 20212 citations

Computing with Noisy Information

Uriel Feige, Prabhakar Raghavan, David Peleg +1 more

This paper studies the depth of noisy decision trees in which each node gives the wrong answer with some constant probability. In the noisy Boolean decision tree model, tight bounds are given on the number of queries to input variables required to compute threshold functions, the parity function and symmetric functions. In the noisy comparison tree model, tight bounds are given on the number of noisy comparisons for searching, sorting, selection and merging. The paper also studies parallel selection and sorting with noisy comparisons, giving tight bounds for several problems.This paper studies the depth of noisy decision trees in which each node gives the wrong answer with some constant probability. In the noisy Boolean decision tree model, tight bounds are given on the number of queries to input variables required to compute threshold functions, the parity function and symmetric functions. In the noisy comparison tree model, tight bounds are given on the number of noisy comparisons for searching, sorting, selection and merging. The paper also studies parallel selection and sorting with noisy comparisons, giving tight bounds for several problems.

SIAM Journal on ComputingOct 1, 1994303 citations

\Me(n \log n)

\Me(n \log n)

We study the query complexity of Boolean functions

f: \{0, 1\}^n \rightarrow \{0, 1\}

in the noisy query model introduced by Feige, Raghavan, Peleg and Upfal [SICOMP 1994]. In this model, an algorithm can adaptively query the bits of an input vector, but each query result is independently flipped with constant probability

p \in (0, 1/2)

; repeated queries are allowed. The noisy query complexity

\mathsf{N}_p(f)

of a function

f

is defined as the minimum expected number of queries needed to compute

f(x)

with error probability at most

1/3

, for the worst case input

x

. We prove a general lower bound on

\mathsf{N}_p(f)

based on degree statistics of certain subgraphs of the Boolean hypercube. This is the first general lower bound beyond those implied by the simple observation that

\mathsf{N}_p(f)

is lower bounded by the randomized query complexity. We show that this recovers (up to a constant factor) most previously known lower bounds on the noisy query complexity of Boolean functions, providing a unified framework for understanding these results and simplifying the proofs in several cases. Furthermore, this resolves in the affirmative an open problem of Gu, Li and Xu [COLT 2025] that

\mathsf{N}_p(f) = Ω(\mathsf{I}(f) \log \mathsf{I}(f))

, where

\mathsf{I}(f)

denotes the total influence of

f

. We also apply our general lower bound to obtain tight bounds on the noisy query complexity for several new functions.

In the noisy query model, the (binary) return value of every query (possibly repeated) is independently flipped with some fixed probability

p \in (0, 1/2)

. In this paper, we obtain tight bounds on the noisy query complexity of several fundamental problems. Our first contribution is to show that any Boolean function with total influence

Ω(n)

has noisy query complexity

Θ(n\log n)

. Previous works often focus on specific problems, and it is of great interest to have a characterization of noisy query complexity for general functions. Our result is the first noisy query complexity lower bound of this generality, beyond what was known for random Boolean functions [Reischuk and Schmeltz, FOCS 1991]. Our second contribution is to prove that Graph Connectivity has noisy query complexity

Θ(n^2 \log n)

. In this problem, the goal is to determine whether an undirected graph is connected using noisy edge queries. While the upper bound can be achieved by a simple algorithm, no non-trivial lower bounds were known prior to this work. Last but not least, we determine the exact number of noisy queries (up to lower order terms) needed to solve the

k

-Threshold problem and the Counting problem. The

k

-Threshold problem asks to decide whether there are at least

k

ones among

n

bits, given noisy query access to the bits. We prove that

(1\pm o(1)) \frac{n\log (\min\{k,n-k+1\}/δ)}{(1-2p)\log \frac{1-p}p}

queries are both sufficient and necessary to achieve error probability

δ= o(1)

. Previously, such a result was only known when

\min\{k,n-k+1\}=o(n)

[Wang, Ghaddar, Zhu and Wang, arXiv 2024]. We also show a similar

(1\pm o(1)) \frac{n\log (\min\{k+1,n-k+1\}/δ)}{(1-2p)\log \frac{1-p}p}

bound for the Counting problem, where one needs to count the number of ones among

n

bits given noisy query access and

k

denotes the answer.

We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on

n

input bits, each of which has approximate Fourier sparsity at most

O(n^3)

and randomized parity decision tree complexity

Θ(n)

. This improves upon the recent work of Chattopadhyay, Mande and Sherif (JACM '20) both qualitatively (in terms of designing a large number of examples) and quantitatively (improving the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).