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Ajtai, Miklós

doi:10.4086/toc.2005.v001a008

Cited by 17 publications

(12 citation statements)

References 7 publications

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“…When s = n(1 + Ω(1)), (1.1) is at best logarithmic. Such logarithmic lower bounds were obtained for m = n 1+Ω (1) by Siegel [23], Theorem 3.1, for computing hash functions. For several settings of parameters [23] also shows that the bound is tight.…”

Section: Introductionmentioning

confidence: 93%

“…It remains to see how to answer queries fast. Because the depth is o(log n), the value at every output gate depends on at most n o (1) wires that were removed, and at most n o (1) input bits. Thus reading the corresponding bits we know the value of the output gate.…”

Section: Bounded Fan-in Circuitsmentioning

confidence: 99%

“…Miltersen, Nisan, Safra, and Wigderson [18] showed that data-structure lower bounds imply lower bounds for branching programs which read each variable few times. However stronger branching-program lower bounds were later proved by Ajtai [1] (see also [3]).…”

Section: Introductionmentioning

confidence: 98%

“…For a streamlined exposition of this lower bound and matching upper bound, in the case of non-adaptive, binary queries see Lecture 18 in [26]. Remarkably, if s = n(1 + Ω(1)) and m = O(s), or if s = n 1+Ω (1) no lower bound is known. Counting arguments (Theorem 8 in [17]) show the existence of functions requiring polynomial time even for space s = m − 1.…”

Section: Introductionmentioning

confidence: 99%

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Viola¹

2019

Theory of Comput.

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Let f : {0, 1} n → {0, 1} m be a function computable by a circuit with unbounded fan-in, arbitrary gates, w wires and depth d. With a very simple argument we show that the m-query problem corresponding to f has data structures with space s = n + r and time (w/r) d , for any r. As a consequence, in the setting where s is close to m a slight improvement on the state of existing data-structure lower bounds would solve long-standing problems in circuit complexity. We also use this connection to obtain a data structure for error-correcting codes which nearly matches the 2007 lower bound by Gál and Miltersen. This data structure can also be made dynamic. Finally we give a problem that requires at least 3 bit probes for m = n O(1) and even s = m/2 − 1. Independent work by Dvir, Golovnev, and Weinstein (2018) and by Corrigan-Gibbs and Kogan (2018) give incomparable connections between data-structure and other types of lower bounds.

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Section: Introductionmentioning

confidence: 93%

Section: Bounded Fan-in Circuitsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Viola¹

2019

Theory of Comput.

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“…The semantic model seemed more difficult, but nevertheless Ajtai [Ajt05] was able to prove an exponential lower bound for semantic read-k-times programs (when k is constant), which was extended by Beame at al. [BSSV03] to randomized branching programs.…”

Section: Related Workmentioning

confidence: 99%

Identity Testing and Lower Bounds for Read- k Oblivious Algebraic Branching Programs

Anderson

Forbes

Saptharishi

et al. 2018

ACM Trans. Comput. Theory

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Read-k oblivious algebraic branching programs are a natural generalization of the wellstudied model of read-once oblivious algebraic branching program (ROABPs). In this work, we give an exponential lower bound of exp(n/k O(k) ) on the width of any read-k oblivious ABP computing some explicit multilinear polynomial f that is computed by a polynomial size depth-3 circuit. We also study the polynomial identity testing (PIT) problem for this model and obtain a white-box subexponential-time PIT algorithm. The algorithm runs in time 2Õ (n 1−1/2 k−1 ) and needs white box access only to know the order in which the variables appear in the ABP.

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Provable Time-Memory Trade-Offs: Symmetric Cryptography Against Memory-Bounded Adversaries

Tessaro

Thiruvengadam

2018

Theory of Cryptography

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Cited by 17 publications

References 7 publications

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Identity Testing and Lower Bounds for Read- k Oblivious Algebraic Branching Programs

Provable Time-Memory Trade-Offs: Symmetric Cryptography Against Memory-Bounded Adversaries

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