  Under the auspices of the Computational Complexity Foundation (CCF)     REPORTS > KEYWORD > MONOTONICITY TESTING:
Reports tagged with Monotonicity testing:
TR13-062 | 18th April 2013
C. Seshadhri, Deeparnab Chakrabarty

#### An optimal lower bound for monotonicity testing over hypergrids

For positive integers $n, d$, consider the hypergrid $[n]^d$ with the coordinate-wise product partial ordering denoted by $\prec$.
A function $f: [n]^d \mapsto \mathbb{N}$ is monotone if $\forall x \prec y$, $f(x) \leq f(y)$.
A function $f$ is $\varepsilon$-far from monotone if at least an $\varepsilon$-fraction of values must ... more >>>

TR15-011 | 22nd January 2015
Subhash Khot, Dor Minzer, Muli Safra

#### On Monotonicity Testing and Boolean Isoperimetric type Theorems

We show a directed and robust analogue of a boolean isoperimetric type theorem of Talagrand. As an application, we
give a monotonicity testing algorithm that makes $\tilde{O}(\sqrt{n}/\epsilon^2)$ non-adaptive queries to a function
$f:\{0,1\}^n \mapsto \{0,1\}$, always accepts a monotone function and rejects a function that is $\epsilon$-far from
being monotone ... more >>>

TR17-159 | 28th October 2017

#### A $o(d) \cdot \text{polylog}~n$ Monotonicity Tester for Boolean Functions over the Hypergrid $[n]^d$

We study monotonicity testing of Boolean functions over the hypergrid $[n]^d$ and design a non-adaptive tester with $1$-sided error whose query complexity is $\tilde{O}(d^{5/6})\cdot \text{poly}(\log n,1/\epsilon)$. Previous to our work, the best known testers had query complexity linear in $d$ but independent of $n$. We improve upon these testers as ... more >>>

TR18-005 | 9th January 2018
C. Seshadhri, Deeparnab Chakrabarty

#### Adaptive Boolean Monotonicity Testing in Total Influence Time

The problem of testing monotonicity
of a Boolean function $f:\{0,1\}^n \to \{0,1\}$ has received much attention
recently. Denoting the proximity parameter by $\varepsilon$, the best tester is the non-adaptive $\widetilde{O}(\sqrt{n}/\varepsilon^2)$ tester
of Khot-Minzer-Safra (FOCS 2015). Let $I(f)$ denote the total influence
of $f$. We give an adaptive tester whose running ... more >>>

TR18-187 | 4th November 2018

#### Domain Reduction for Monotonicity Testing: A $o(d)$ Tester for Boolean Functions on Hypergrids

Revisions: 4

Testing monotonicity of Boolean functions over the hypergrid, $f:[n]^d \to \{0,1\}$, is a classic problem in property testing. When the range is real-valued, there are $\Theta(d\log n)$-query testers and this is tight. In contrast, the Boolean range qualitatively differs in two ways:
(1) Independence of $n$: There are testers ... more >>>

TR18-196 | 19th November 2018
Omri Ben-Eliezer

#### Testing local properties of arrays

We study testing of local properties in one-dimensional and multi-dimensional arrays. A property of $d$-dimensional arrays $f:[n]^d \to \Sigma$ is $k$-local if it can be defined by a family of $k \times \ldots \times k$ forbidden consecutive patterns. This definition captures numerous interesting properties. For example, monotonicity, Lipschitz continuity and ... more >>>

TR19-134 | 4th October 2019
Omri Ben-Eliezer, Clement Canonne, Shoham Letzter, Erik Waingarten

#### Finding monotone patterns in sublinear time

We study the problem of finding monotone subsequences in an array from the viewpoint of sublinear algorithms. For fixed $k \in \mathbb{N}$ and $\varepsilon > 0$, we show that the non-adaptive query complexity of finding a length-$k$ monotone subsequence of $f \colon [n] \to \mathbb{R}$, assuming that $f$ is $\varepsilon$-far ... more >>>

TR19-163 | 16th November 2019
Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, Erik Waingarten

Revisions: 1

We design a nonadaptive algorithm that, given a Boolean function $f\colon \{0,1\}^n \to \{0,1\}$ which is $\alpha$-far from monotone, makes poly$(n, 1/\alpha)$ queries and returns an estimate that, with high probability, is an $\widetilde{O}(\sqrt{n})$-approximation to the distance of $f$ to monotonicity. Furthermore, we show that for any constant $\kappa > ... more >>> TR20-174 | 18th November 2020 Hadley Black, Iden Kalemaj, Sofya Raskhodnikova #### Isoperimetric Inequalities for Real-Valued Functions with Applications to Monotonicity Testing We generalize the celebrated isoperimetric inequality of Khot, Minzer, and Safra (SICOMP 2018) for Boolean functions to the case of real-valued functions$f \colon \{0,1\}^d\to\mathbb{R}$. Our main tool in the proof of the generalized inequality is a new Boolean decomposition that represents every real-valued function$f\$ over an arbitrary partially ... more >>>

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