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Belovs, Aleksandrs; Blais, Eric

doi:10.4086/toc.2015.v011a016

Cited by 18 publications

(4 citation statements)

References 24 publications

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“…Now let us turn to the lower bounds. Ergün et al [12] proved a lower bound 1 of Ω(log n) for testing monotonicity on the line in the so-called comparison-based model. In this model, each query of the tester may depend only on the order relations between the responses to the previous queries, but not on the values of the responses themselves.…”

Section: Introductionmentioning

confidence: 99%

A polynomial lower bound for testing monotonicity

Belovs

Blais

2016

Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing

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We show that every algorithm for testing n-variate Boolean functions for monotonicity has query complexity Ω(n 1/4 ). All previous lower bounds for this problem were designed for nonadaptive algorithms and, as a result, the best previous lower bound for general (possibly adaptive) monotonicity testers was only Ω(log n). Combined with the query complexity of the non-adaptive monotonicity tester of Khot, Minzer, and Safra (FOCS 2015), our lower bound shows that adaptivity can result in at most a quadratic reduction in the query complexity for testing monotonicity.By contrast, we show that there is an exponential gap between the query complexity of adaptive and non-adaptive algorithms for testing regular linear threshold functions (LTFs) for monotonicity. Chen, De, Servedio, and Tan (STOC 2015) recently showed that non-adaptive algorithms require almost Ω(n 1/2 ) queries for this task. We introduce a new adaptive monotonicity testing algorithm which has query complexity O(log n) when the input is a regular LTF.

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Section: Introductionmentioning

confidence: 99%

A polynomial lower bound for testing monotonicity

Belovs

Blais

2016

Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing

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“…The negative-weighted adversary bound has been used to prove lower bounds [22,19,20], but more frequently to prove upper bounds, in particular using the learning graph approach [12]. For instance, the adversary bound (sometimes in the equivalent form of span programs) was used to construct quantum algorithms for formula evaluation [55,54,61], finding subgraphs [43,18,42], k-distinctness problem [11], and in learning and property testing [14,16].…”

Section: Adversary Boundmentioning

confidence: 99%

One-Way Ticket to Las Vegas and the Quantum Adversary

Belovs¹,

Yolcu²

2023

Preprint

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We propose a new definition of quantum Las Vegas query complexity. We show that it is exactly equal to the quantum adversary bound. This is achieved by a new and very simple way of transforming a feasible solution to the adversary optimisation problem into a quantum query algorithm. This allows us to generalise the bound to include unidirectional access, multiple input oracles, and input oracles that are not unitary. As an application, we demonstrate a separation between unidirectional and bidirectional access to an input oracle for a rather natural unitary permutation inversion problem.

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“…where we used a r c < a c r , (1 − 2 −κ ) −1 ≤ 2 and a r 2 κ−1 ≤ 1 2 to get the first, second and third inequalities, respectively. Using the union bound over all r ∈ [d] and grouping according to n , we get 8) where the second last inequality follows from Equations (3.6) and (3.7), and the last inequality holds because if coordinate r is included in the count n , then 2 −1 < a r ≤ 2 .…”

Section: Bad Eventsmentioning

confidence: 99%

“…Testing various properties of functions, including monotonicity (see, e. g., [38,52,33,34,47,36,53,35,41,1,42,5,16,12,19,22,17,11,23,24,21,27,26,45,8,9,32,49,29,7,13,25,14,50] and recent surveys [54,55,20]), the Lipschitz property [43,22,31,17,2], bounded-derivative properties [21], linearity [18,6,10,44,56], submodularity [51,58,60,…”

Section: Introductionmentioning

confidence: 99%

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Baleshzar¹,

Chakrabarty²,

Pallavoor³

et al. 2020

Theory of Comput.

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We study the problem of testing unateness of functions f : {0, 1} d → R. A function f : {0, 1} d → R is unate if for every coordinate i ∈ [d], the function is either nonincreasing in the i th coordinate or nondecreasing in the i th coordinate. We give an O((d/ε) • log(d/ε))-query nonadaptive tester and an O(d/ε)-query adaptive tester and show that both testers are optimal for a fixed distance parameter ε. Previously known unateness testers worked only for Boolean functions, and their query complexity had worse dependence on the dimension d both for the adaptive and the nonadaptive case. Moreover, no lower bounds for testing unateness were known. (Concurrent work by Chen et al. (STOC'17) proved an Ω(d/ log 2 d) lower bound on the nonadaptive query complexity of testing unateness of Boolean functions.) We also generalize our results to obtain optimal unateness testers for functions f : [n] d → R.

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Cited by 18 publications

References 24 publications

A polynomial lower bound for testing monotonicity

A polynomial lower bound for testing monotonicity

One-Way Ticket to Las Vegas and the Quantum Adversary

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