Separation Between Deterministic and Randomized Query Complexity

Mukhopadhyay, Sagnik; Radhakrishnan, Jaikumar; Sanyal, Swagato

doi:10.1137/17m1124115

Cited by 6 publications

(10 citation statements)

References 12 publications

(6 reference statements)

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“…Using the Göös-Pitassi-Watson function, it is already possible to give a larger separation between randomized and deterministic query complexity than previously known. For instance, Mukhopadhyay and Sanyal [17], independently from our work, obtained separations R(f ) = O( D(f )) and R 0 (f ) = O(D(f ) 3/4 ). However, these algorithms are rather complicated, and it is not known whether this function can realize an optimal separation between R 0 (f ) and D(f ).…”

Section: Our Technique and Pointer Functionsmentioning

confidence: 62%

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Separations in query complexity based on pointer functions

Ambainis

Balodis

Belovs

et al. 2016

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

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In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total boolean function is given by the function f on n = 2 k bits defined by a complete binary tree of NAND gates of depth k, which achieves R 0 (f ) = O(D(f ) 0.7537... ). We show this is false by giving an example of a total boolean function f on n bits whose deterministic query complexity is Ω(n/ log(n)) while its zero-error randomized query complexity is O( √ n). We further show that the quantum query complexity of the same function is O(n 1/4 ), giving the first example of a total function with a super-quadratic gap between its quantum and deterministic query complexities.We also construct a total boolean function g on n variables that has zero-error randomized query complexity Ω(n/ log(n)) and bounded-error randomized query complexity R(g) = O( √ n). This is the first super-linear separation between these two complexity measures. The exact quantum query complexity of the same function is Q E (g) = O( √ n). These two functions show that the relations D(f ) = O(R 1 (f ) 2 ) and R 0 (f ) = O(R(f ) 2 ) are optimal, up to poly-logarithmic factors. Further variations of these functions give additional separations between other query complexity measures: a cubic separation between Q and R 0 , a 3/2-power separation between Q E and R, and a 4th power separation between approximate degree and bounded-error randomized query complexity.All of these examples are variants of a function recently introduced by Göös, Pitassi, and Watson which they used to separate the unambiguous 1-certificate complexity from deterministic query complexity and to resolve the famous Clique versus Independent Set problem in communication complexity.

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Section: Our Technique and Pointer Functionsmentioning

confidence: 62%

“…As the zeroes are arranged in a path, the algorithm can thus eliminate half of the potential marked columns on average. This fact is exploited by the algorithm of Mukhopadhyay and Sanyal [17].…”

Section: Our Technique and Pointer Functionsmentioning

confidence: 99%

Separations in query complexity based on pointer functions

Ambainis

Balodis

Belovs

et al. 2016

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

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“…To show a gap between D(f ) and R 2 (f ), it suffices to show that a randomized algorithm can find this certificate faster than a deterministic algorithm. For that, we set N = M and consider the following randomized algorithm (due to Mukhopadhyay and Sanyal [34]):…”

Section: There Exists a Total Boolean Functionmentioning

confidence: 99%

Understanding Quantum Algorithms via Query Complexity

Ambainis

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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Query complexity is a model of computation in which we have to compute a function f (x1, . . . , xN ) of variables xi which can be accessed via queries. The complexity of an algorithm is measured by the number of queries that it makes. Query complexity is widely used for studying quantum algorithms, for two reasons. First, it includes many of the known quantum algorithms (including Grover's quantum search and a key subroutine of Shor's factoring algorithm). Second, one can prove lower bounds on the query complexity, bounding the possible quantum advantage. In the last few years, there have been major advances on several longstanding problems in the query complexity. In this talk, we survey these results and related work, including:• the biggest quantum-vs-classical gap for partial functions (a problem solvable with 1 query quantumly but requiring Ω( √ N ) queries classically);• the biggest quantum-vs-determistic and quantum-vs-probabilistic gaps for total functions (for example, a problem solvable with M queries quantumly but requiringΩ(M 2.5 ) queries probabilistically);• the biggest probabilistic-vs-deterministic gap for total functions (a problem solvable with M queries probabilistically but requiringΩ(M 2 ) queries deterministically);• the bounds on the gap that can be achieved for subclasses of functions (for example, symmetric functions);• the connections between query algorithms and approximations by low-degree polynomials.

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“…For any boolean function f on n variables, R 0 (f ) = Ω(D(f ) 0.753... ). This conjecture was recently refuted independently by Ambainis et al [2] and Mukhopadhyay and Sanyal [7]. Both works based their result on the pointer function introduced by Göös, Pitassi and Watson [5], who used this function to show a separation between deterministic decision tree complexity and unambiguous non-deterministic decision tree complexity.…”

Section: Conjecture 1 ([9]mentioning

confidence: 99%

“…Mukhopadhyay and Sanyal [7] used GPW s×s to obtain the following refutation of Conjecture 1: R 0 (GPW s×s ) = O(s 1.5 ) while D(GPW s×s ) = Ω(s 2 ). While this shows that GPW s×s witnesses a wider separation between deterministic and zero-error randomized query complexities than conjectured, the separation shown is not the widest possible for a Boolean function.…”

Section: Conjecture 1 ([9]mentioning

confidence: 99%

The zero-error randomized query complexity of the pointer function

Radhakrishnan,

Sanyal

2016

Preprint

Self Cite

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The pointer function of Göös, Pitassi and Watson [5] and its variants have recently been used to prove separation results among various measures of complexity such as deterministic, randomized and quantum query complexities, exact and approximate polynomial degrees, etc. In particular, the widest possible (quadratic) separations between deterministic and zero-error randomized query complexity, as well as between bounded-error and zero-error randomized query complexity, have been obtained by considering variants [2] of this pointer function.However, as was pointed out in [2], the precise zero-error complexity of the original pointer function was not known. We show a lower bound of Ω(n 3/4 ) on the zero-error randomized query complexity of the pointer function on Θ(n log n) bits; since an O(n 3/4 ) upper bound is already known [7], our lower bound is optimal up to a factor of polylog n.

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Separation Between Deterministic and Randomized Query Complexity

Abstract: We show that there exists a Boolean function F which observes the following separations among deterministic query complexity (

Cited by 6 publications

References 12 publications

Separations in query complexity based on pointer functions

Separations in query complexity based on pointer functions

Understanding Quantum Algorithms via Query Complexity

The zero-error randomized query complexity of the pointer function

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