4 Search Results for "Huang, Xuangui"


Document
RANDOM
Affine Extractors and AC0-Parity

Authors: Xuangui Huang, Peter Ivanov, and Emanuele Viola

Published in: LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)


Abstract
We study a simple and general template for constructing affine extractors by composing a linear transformation with resilient functions. Using this we show that good affine extractors can be computed by non-explicit circuits of various types, including AC0-Xor circuits: AC0 circuits with a layer of parity gates at the input. We also show that one-sided extractors can be computed by small DNF-Xor circuits, and separate these circuits from other well-studied classes. As a further motivation for studying DNF-Xor circuits we show that if they can approximate inner product then small AC0-Xor circuits can compute it exactly - a long-standing open problem.

Cite as

Xuangui Huang, Peter Ivanov, and Emanuele Viola. Affine Extractors and AC0-Parity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{huang_et_al:LIPIcs.APPROX/RANDOM.2022.9,
  author =	{Huang, Xuangui and Ivanov, Peter and Viola, Emanuele},
  title =	{{Affine Extractors and AC0-Parity}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{9:1--9:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.9},
  URN =		{urn:nbn:de:0030-drops-171313},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.9},
  annote =	{Keywords: affine extractor, resilient function, constant-depth circuit, parity gate, inner product}
}
Document
Space Hardness of Solving Structured Linear Systems

Authors: Xuangui Huang

Published in: LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)


Abstract
Space-efficient Laplacian solvers are closely related to derandomization of space-bound randomized computations. We show that if the probabilistic logarithmic-space solver or the deterministic nearly logarithmic-space solver for undirected Laplacian matrices can be extended to solve slightly larger subclasses of linear systems, then they can be used to solve all linear systems with similar space complexity. Previously Kyng and Zhang [Rasmus Kyng and Peng Zhang, 2017] proved such results in the time complexity setting using reductions between approximate solvers. We prove that their reductions can be implemented using constant-depth, polynomial-size threshold circuits.

Cite as

Xuangui Huang. Space Hardness of Solving Structured Linear Systems. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 56:1-56:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{huang:LIPIcs.ISAAC.2020.56,
  author =	{Huang, Xuangui},
  title =	{{Space Hardness of Solving Structured Linear Systems}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{56:1--56:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Cao, Yixin and Cheng, Siu-Wing and Li, Minming},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.56},
  URN =		{urn:nbn:de:0030-drops-134001},
  doi =		{10.4230/LIPIcs.ISAAC.2020.56},
  annote =	{Keywords: linear system solver, logarithmic space, threshold circuit}
}
Document
RANDOM
Approximate Degree, Secret Sharing, and Concentration Phenomena

Authors: Andrej Bogdanov, Nikhil S. Mande, Justin Thaler, and Christopher Williamson

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)


Abstract
The epsilon-approximate degree deg~_epsilon(f) of a Boolean function f is the least degree of a real-valued polynomial that approximates f pointwise to within epsilon. A sound and complete certificate for approximate degree being at least k is a pair of probability distributions, also known as a dual polynomial, that are perfectly k-wise indistinguishable, but are distinguishable by f with advantage 1 - epsilon. Our contributions are: - We give a simple, explicit new construction of a dual polynomial for the AND function on n bits, certifying that its epsilon-approximate degree is Omega (sqrt{n log 1/epsilon}). This construction is the first to extend to the notion of weighted degree, and yields the first explicit certificate that the 1/3-approximate degree of any (possibly unbalanced) read-once DNF is Omega(sqrt{n}). It draws a novel connection between the approximate degree of AND and anti-concentration of the Binomial distribution. - We show that any pair of symmetric distributions on n-bit strings that are perfectly k-wise indistinguishable are also statistically K-wise indistinguishable with at most K^{3/2} * exp (-Omega (k^2/K)) error for all k < K <= n/64. This bound is essentially tight, and implies that any symmetric function f is a reconstruction function with constant advantage for a ramp secret sharing scheme that is secure against size-K coalitions with statistical error K^{3/2} * exp (-Omega (deg~_{1/3}(f)^2/K)) for all values of K up to n/64 simultaneously. Previous secret sharing schemes required that K be determined in advance, and only worked for f=AND. Our analysis draws another new connection between approximate degree and concentration phenomena. As a corollary of this result, we show that for any d <= n/64, any degree d polynomial approximating a symmetric function f to error 1/3 must have coefficients of l_1-norm at least K^{-3/2} * exp ({Omega (deg~_{1/3}(f)^2/d)}). We also show this bound is essentially tight for any d > deg~_{1/3}(f). These upper and lower bounds were also previously only known in the case f=AND.

Cite as

Andrej Bogdanov, Nikhil S. Mande, Justin Thaler, and Christopher Williamson. Approximate Degree, Secret Sharing, and Concentration Phenomena. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 71:1-71:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{bogdanov_et_al:LIPIcs.APPROX-RANDOM.2019.71,
  author =	{Bogdanov, Andrej and Mande, Nikhil S. and Thaler, Justin and Williamson, Christopher},
  title =	{{Approximate Degree, Secret Sharing, and Concentration Phenomena}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{71:1--71:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.71},
  URN =		{urn:nbn:de:0030-drops-112869},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.71},
  annote =	{Keywords: approximate degree, dual polynomial, pseudorandomness, polynomial approximation, secret sharing}
}
Document
Slicewise Definability in First-Order Logic with Bounded Quantifier Rank

Authors: Yijia Chen, Jörg Flum, and Xuangui Huang

Published in: LIPIcs, Volume 82, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)


Abstract
For every natural number q let FO_q denote the class of sentences of first-order logic FO of quantifier rank at most q. If a graph property can be defined in FO_q, then it can be decided in time O(n^q). Thus, minimizing q has favorable algorithmic consequences. Many graph properties amount to the existence of a certain set of vertices of size k. Usually this can only be expressed by a sentence of quantifier rank at least k. We use the color coding method to demonstrate that some (hyper)graph problems can be defined in FO_q where q is independent of k. This property of a graph problem is equivalent to the question of whether the corresponding parameterized problem is in the class para-AC^0. It is crucial for our results that the FO-sentences have access to built-in addition and multiplication (and constants for an initial segment of natural numbers whose length depends only on k). It is known that then FO corresponds to the circuit complexity class uniform AC^0. We explore the connection between the quantifier rank of FO-sentences and the depth of AC^0-circuits, and prove that FO_q is strictly contained in FO_{q+1} for structures with built-in addition and multiplication.

Cite as

Yijia Chen, Jörg Flum, and Xuangui Huang. Slicewise Definability in First-Order Logic with Bounded Quantifier Rank. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{chen_et_al:LIPIcs.CSL.2017.19,
  author =	{Chen, Yijia and Flum, J\"{o}rg and Huang, Xuangui},
  title =	{{Slicewise Definability in First-Order Logic with Bounded Quantifier Rank}},
  booktitle =	{26th EACSL Annual Conference on Computer Science Logic (CSL 2017)},
  pages =	{19:1--19:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-045-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{82},
  editor =	{Goranko, Valentin and Dam, Mads},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2017.19},
  URN =		{urn:nbn:de:0030-drops-76742},
  doi =		{10.4230/LIPIcs.CSL.2017.19},
  annote =	{Keywords: first-order logic, quantifier rank, parameterized AC^0, circuit depth}
}
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