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ACM Symposium on Theory of Computing (STOC)

A o(n) monotonicity tester for boolean functions over the hypercube

Deeparnab Chakrabarty, Comandur Seshadhri

Published May 28, 2013• 30 citations
Publisher

Abstract

Given oracle access to a Boolean function f:{0,1}n -> {0,1}, we design a randomized tester that takes as input a parameter ε>0, and outputs Yes if the function is monotonically non-increasing, and outputs No with probability >2/3, if the function is ε-far from being monotone, that is, f needs to be modified at ε-fraction of the points to make it monotone. Our non-adaptive, one-sided tester makes ~O(n5/6ε-5/3) queries to the oracle.

DOI
10.1145/2488608.2488660

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Given oracle access to a Boolean function f:{0,1}n -> {0,1}, we design a randomized tester that takes as input a parameter ε>0, and outputs Yes if the function is monotonically non-increasing, and outputs No with probability >2/3, if the function is ε-far from being monotone, that is, f needs to be modified at ε-fraction of the points to make it monotone. Our non-adaptive, one-sided tester makes ~O(n5/6ε-5/3) queries to the oracle.
We show a procedure that, given oracle access to a function f ⁣:{0,1}n→{0,1}f\colon \{0,1\}^n\to\{0,1\}f:{0,1}n→{0,1}, produces oracle access to a function f′ ⁣:{0,1}n′→{0,1}f'\colon \{0,1\}^{n'}\to\{0,1\}f′:{0,1}n′→{0,1} such that if fff is monotone, then f′f'f′ is monotone, and if fff is ε\varepsilonε-far from monotone, then f′f'f′ is Ω(1)Ω(1)Ω(1)-far from monotone. Moreover, n′≤n2O(1/ε)n' \leq n 2^{O(1/\varepsilon)}n′≤n2O(1/ε) and each oracle query to f′f'f′ can be answered by making 2O(1/ε)2^{O(1/\varepsilon)}2O(1/ε) oracle queries to fff. Our lemma is motivated by a recent result of [Chen, Chen, Cui, Pires, Stockwell, arXiv:2511.04558], who showed that for all c>0c>0c>0 there exists εc>0\varepsilon_c>0εc​>0, such that any (even two-sided, adaptive) algorithm distinguishing between monotone functions and εc\varepsilon_cεc​-far from monotone functions, requires Ω(n1/2−c)Ω(n^{1/2-c})Ω(n1/2−c) queries. Combining our lemma with their result implies a similar result, except that the distance from monotonicity is an absolute constant ε>0\varepsilon>0ε>0, and the lower bound is Ω(n1/2−o(1))Ω(n^{1/2-o(1)})Ω(n1/2−o(1)) queries.
In this note, we develop a bounded-error quantum algorithm that makes O~(n1/4ε−1/2)\tilde O(n^{1/4}\varepsilon^{-1/2})O~(n1/4ε−1/2) queries to a Boolean function fff, accepts a monotone function, and rejects a function that is ε\varepsilonε-far from being monotone. This gives a super-quadratic improvement compared to the best known randomized algorithm for all ε=o(1)\varepsilon = o(1)ε=o(1). The improvement is cubic when ε=1/n\varepsilon = 1/\sqrt{n}ε=1/n​.
The noise sensitivity of a Boolean function f: {0,1}^n - > {0,1} is one of its fundamental properties. For noise parameter delta, the noise sensitivity is denoted as NS_{delta}[f]. This quantity is defined as follows: First, pick x = (x_1,...,x_n) uniformly at random from {0,1}^n, then pick z by flipping each x_i independently with probability delta. NS_{delta}[f] is defined to equal Pr [f(x) != f(z)]. Much of the existing literature on noise sensitivity explores the following two directions: (1) Showing that functions with low noise-sensitivity are structured in certain ways. (2) Mathematically showing that certain classes of functions have low noise sensitivity. Combined, these two research directions show that certain classes of functions have low noise sensitivity and therefore have useful structure. The fundamental importance of noise sensitivity, together with this wealth of structural results, motivates the algorithmic question of approximating NS_{delta}[f] given an oracle access to the function f. We show that the standard sampling approach is essentially optimal for general Boolean functions. Therefore, we focus on estimating the noise sensitivity of monotone functions, which form an important subclass of Boolean functions, since many functions of interest are either monotone or can be simply transformed into a monotone function (for example the class of unate functions consists of all the functions that can be made monotone by reorienting some of their coordinates [O'Donnell, 2014]). Specifically, we study the algorithmic problem of approximating NS_{delta}[f] for monotone f, given the promise that NS_{delta}[f] >= 1/n^{C} for constant C, and for delta in the range 1/n <= delta <= 1/2. For such f and delta, we give a randomized algorithm performing O((min(1,sqrt{n} delta log^{1.5} n))/(NS_{delta}[f]) poly (1/epsilon)) queries and approximating NS_{delta}[f] to within a multiplicative factor of (1 +/- epsilon). Given the same constraints on f and delta, we also prove a lower bound of Omega((min(1,sqrt{n} delta))/(NS_{delta}[f] * n^{xi})) on the query complexity of any algorithm that approximates NS_{delta}[f] to within any constant factor, where xi can be any positive constant. Thus, our algorithm’s query complexity is close to optimal in terms of its dependence on n. We introduce a novel descending-ascending view of noise sensitivity, and use it as a central tool for the analysis of our algorithm. To prove lower bounds on query complexity, we develop a technique that reduces computational questions about query complexity to combinatorial questions about the existence of "thin" functions with certain properties. The existence of such "thin" functions is proved using the probabilistic method. These techniques also yield new lower bounds on the query complexity of approximating other fundamental properties of Boolean functions: the total influence and the bias.
Let f:{-1,1}^n -> R be a real function on the hypercube, given by its discrete Fourier expansion, or, equivalently, represented as a multilinear polynomial. We say that it is Boolean if its image is in {-1,1}. We show that every function on the hypercube with a sparse Fourier expansion must either be Boolean or far from Boolean. In particular, we show that a multilinear polynomial with at most k terms must either be Boolean, or output values different than -1 or 1 for a fraction of at least 2/(k+2)^2 of its domain. It follows that given oracle access to f, together with the guarantee that its representation as a multilinear polynomial has at most k terms, one can test Booleanity using O(k^2) queries. We show an Ω(k) queries lower bound for this problem. Our proof crucially uses Hirschman's entropic version of Heisenberg's uncertainty principle.
We define and study a new type of quantum oracle, the quantum conditional oracle, which provides oracle access to the conditional probabilities associated with an underlying distribution. Amongst other properties, we (a) obtain highly efficient quantum algorithms for identity testing, equivalence testing and uniformity testing of probability distributions; (b) study the power of these oracles for testing properties of boolean functions, and obtain an algorithm for checking whether an n-input m-output boolean function is balanced or e-far from balanced; and (c) give an algorithm, requiring O˜(n/e) queries, for testing whether an n-dimensional quantum state is maximally mixed or not.