Document

**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Recently, there has been exciting progress in understanding the complexity of distributions. Here, the goal is to quantify the resources required to generate (or sample) a distribution. Proving lower bounds in this new setting is more challenging than in the classical setting, and has yielded interesting new techniques and surprising applications. In this work, we initiate a study of the complexity of sampling with limited memory, and obtain the first nontrivial sampling lower bounds against oblivious read-once branching programs (ROBPs).
In our first main result, we show that any distribution sampled by an ROBP of width 2^{Ω(n)} has statistical distance 1-2^{-Ω(n)} from any distribution that is uniform over a good code. More generally, we obtain sampling lower bounds for any list decodable code, which are nearly tight. Previously, such a result was only known for sampling in AC⁰ (Lovett and Viola, CCC'11; Beck, Impagliazzo and Lovett, FOCS'12). As an application of our result, a known connection implies new data structure lower bounds for storing codewords.
In our second main result, we prove a direct product theorem for sampling with ROBPs. Previously, no direct product theorems were known for the task of sampling, for any computational model. A key ingredient in our proof is a simple new lemma about amplifying statistical distance between sequences of somewhat-dependent random variables. Using this lemma, we also obtain a simple new proof of a known lower bound for sampling disjoint sets using two-party communication protocols (Göös and Watson, RANDOM'19).

Eshan Chattopadhyay, Jesse Goodman, and David Zuckerman. The Space Complexity of Sampling. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 40:1-40:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chattopadhyay_et_al:LIPIcs.ITCS.2022.40, author = {Chattopadhyay, Eshan and Goodman, Jesse and Zuckerman, David}, title = {{The Space Complexity of Sampling}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {40:1--40:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.40}, URN = {urn:nbn:de:0030-drops-156366}, doi = {10.4230/LIPIcs.ITCS.2022.40}, annote = {Keywords: Complexity of distributions, complexity of sampling, extractors, list decodable codes, lower bounds, read-once branching programs, small-space computation} }

Document

**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

We present a randomness efficient version of the linear noise operator T_ρ from boolean function analysis by constructing a sparse linear operator on the space of boolean functions {0,1}ⁿ → {0,1} with similar eigenvalue profile to T_ρ. The linear operator we construct is a direct consequence of a generalization of ε-biased sets to the product distribution 𝒟_p on {0,1}ⁿ where the marginal of each coordinate is p = 1/2-1/2ρ. Such a generalization is a small support distribution that fools linear tests when the input of the test comes from 𝒟_p instead of the uniform distribution. We give an explicit construction of such a distribution that requires log n + O_{p}(log log n + log1/(ε)) bits of uniform randomness to sample from, where the p subscript hides O(log² 1/p) factors. When p and ε are constant, this yields a support size nearly linear in n, whereas previous best known constructions only guarantee a size of poly(n). Furthermore, our construction implies an explicitly constructible "sparse" noisy hypercube graph that is a small set expander.

Dana Moshkovitz, Justin Oh, and David Zuckerman. Randomness Efficient Noise Stability and Generalized Small Bias Sets. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{moshkovitz_et_al:LIPIcs.FSTTCS.2020.31, author = {Moshkovitz, Dana and Oh, Justin and Zuckerman, David}, title = {{Randomness Efficient Noise Stability and Generalized Small Bias Sets}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {31:1--31:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.31}, URN = {urn:nbn:de:0030-drops-132721}, doi = {10.4230/LIPIcs.FSTTCS.2020.31}, annote = {Keywords: pseudorandomness, derandomization, epsilon biased sets, noise stability} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

We give a deterministic, nearly logarithmic-space algorithm for mild spectral sparsification of undirected graphs. Given a weighted, undirected graph G on n vertices described by a binary string of length N, an integer k ≤ log n and an error parameter ε > 0, our algorithm runs in space Õ(k log(N w_max/w_min)) where w_max and w_min are the maximum and minimum edge weights in G, and produces a weighted graph H with Õ(n^(1+2/k)/ε²) edges that spectrally approximates G, in the sense of Spielmen and Teng [Spielman and Teng, 2004], up to an error of ε.
Our algorithm is based on a new bounded-independence analysis of Spielman and Srivastava’s effective resistance based edge sampling algorithm [Spielman and Srivastava, 2011] and uses results from recent work on space-bounded Laplacian solvers [Jack Murtagh et al., 2017]. In particular, we demonstrate an inherent tradeoff (via upper and lower bounds) between the amount of (bounded) independence used in the edge sampling algorithm, denoted by k above, and the resulting sparsity that can be achieved.

Dean Doron, Jack Murtagh, Salil Vadhan, and David Zuckerman. Spectral Sparsification via Bounded-Independence Sampling. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 39:1-39:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{doron_et_al:LIPIcs.ICALP.2020.39, author = {Doron, Dean and Murtagh, Jack and Vadhan, Salil and Zuckerman, David}, title = {{Spectral Sparsification via Bounded-Independence Sampling}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {39:1--39:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.39}, URN = {urn:nbn:de:0030-drops-124462}, doi = {10.4230/LIPIcs.ICALP.2020.39}, annote = {Keywords: Spectral sparsification, Derandomization, Space complexity} }

Document

RANDOM

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

We study the task of seedless randomness extraction from recognizable sources, which are uniform distributions over sets of the form {x : f(x) = 1} for functions f in some specified class C. We give two simple methods for constructing seedless extractors for C-recognizable sources.
Our first method shows that if C admits XOR amplification, then we can construct a seedless extractor for C-recognizable sources by using a mildly hard function for C as a black box. By exploiting this reduction, we give polynomial-time, seedless randomness extractors for three natural recognizable sources: (1) constant-degree algebraic sources over any prime field, where constant-degree algebraic sources are uniform distributions over the set of zeros of a system of constant degree polynomials; (2) sources recognizable by randomized multiparty communication protocols of cn bits, where c>0 is a small enough constant; (3) halfspace sources, or sources recognizable by linear threshold functions. In particular, the new extractor for each of these three sources has linear output length and exponentially small error for min-entropy k >= (1-alpha)n, where alpha>0 is a small enough constant.
Our second method shows that a seed-extending pseudorandom generator with exponentially small error for C yields an extractor with exponentially small error for C-recognizable sources, improving a reduction by Kinne, Melkebeek, and Shaltiel [Kinne et al., 2012]. Using the hardness of the parity function against AC^0 [Håstad, 1987], we significantly improve Shaltiel’s extractor [Shaltiel, 2011] for AC^0-recognizable sources. Finally, assuming sufficiently strong one-way permutations, we construct seedless extractors for sources recognizable by BPP algorithms, and these extractors run in quasi-polynomial time.

Fu Li and David Zuckerman. Improved Extractors for Recognizable and Algebraic Sources. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 72:1-72:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{li_et_al:LIPIcs.APPROX-RANDOM.2019.72, author = {Li, Fu and Zuckerman, David}, title = {{Improved Extractors for Recognizable and Algebraic Sources}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {72:1--72:22}, 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.2019.72}, URN = {urn:nbn:de:0030-drops-112873}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.72}, annote = {Keywords: Extractor, Pseudorandomness} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

The seminal result of Kahn, Kalai and Linial shows that a coalition of O(n/(log n)) players can bias the outcome of any Boolean function {0,1}^n -> {0,1} with respect to the uniform measure. We extend their result to arbitrary product measures on {0,1}^n, by combining their argument with a completely different argument that handles very biased input bits.
We view this result as a step towards proving a conjecture of Friedgut, which states that Boolean functions on the continuous cube [0,1]^n (or, equivalently, on {1,...,n}^n) can be biased using coalitions of o(n) players. This is the first step taken in this direction since Friedgut proposed the conjecture in 2004.
Russell, Saks and Zuckerman extended the result of Kahn, Kalai and Linial to multi-round protocols, showing that when the number of rounds is o(log^* n), a coalition of o(n) players can bias the outcome with respect to the uniform measure. We extend this result as well to arbitrary product measures on {0,1}^n.
The argument of Russell et al. relies on the fact that a coalition of o(n) players can boost the expectation of any Boolean function from epsilon to 1-epsilon with respect to the uniform measure. This fails for general product distributions, as the example of the AND function with respect to mu_{1-1/n} shows. Instead, we use a novel boosting argument alongside a generalization of our first result to arbitrary finite ranges.

Yuval Filmus, Lianna Hambardzumyan, Hamed Hatami, Pooya Hatami, and David Zuckerman. Biasing Boolean Functions and Collective Coin-Flipping Protocols over Arbitrary Product Distributions. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 58:1-58:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{filmus_et_al:LIPIcs.ICALP.2019.58, author = {Filmus, Yuval and Hambardzumyan, Lianna and Hatami, Hamed and Hatami, Pooya and Zuckerman, David}, title = {{Biasing Boolean Functions and Collective Coin-Flipping Protocols over Arbitrary Product Distributions}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {58:1--58:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.58}, URN = {urn:nbn:de:0030-drops-106340}, doi = {10.4230/LIPIcs.ICALP.2019.58}, annote = {Keywords: Boolean function analysis, coin flipping} }

Document

**Published in:** LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

We study how to extract randomness from a C-interleaved source, that is, a source comprised of C independent sources whose bits or symbols are interleaved. We describe a simple approach for constructing such extractors that yields:
(1) For some delta>0, c>0, explicit extractors for 2-interleaved sources on {0,1}^{2n} when one source has min-entropy at least (1-delta)*n and the other has min-entropy at least c*log(n). The best previous construction, by Raz and Yehudayoff, worked only when both sources had entropy rate 1-delta.
(2) For some c>0 and any large enough prime p, explicit extractors for 2-interleaved sources on [p]^{2n} when one source has min-entropy rate at least .51 and the other source has min-entropy rate at least (c*log(n))/n.
We use these to obtain the following applications:
(a) We introduce the class of any-order-small-space sources, generalizing the class of small-space sources studied by Kamp et al.. We construct extractors for such sources with min-entropy rate close to 1/2. Using the Raz-Yehudayoff construction would require entropy rate close to 1.
(b) For any large enough prime p, we exhibit an explicit function f:[p]^{2n} -> {0,1} such that the randomized best-partition communication complexity of f with error 1/2-2^{-Omega(n)} is at least .24*n*log(p). Previously this was known only for a tiny constant instead of .24, for p=2 by by Raz and Yehudayoff.
We introduce non-malleable extractors in the interleaved model. For any large enough prime p, we give an explicit construction of a weak-seeded non-malleable extractor for sources over [p]^n with min-entropy rate .51. Nothing was known previously, even for almost full min-entropy.

Eshan Chattopadhyay and David Zuckerman. New Extractors for Interleaved Sources. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 7:1-7:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chattopadhyay_et_al:LIPIcs.CCC.2016.7, author = {Chattopadhyay, Eshan and Zuckerman, David}, title = {{New Extractors for Interleaved Sources}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {7:1--7:28}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-008-8}, ISSN = {1868-8969}, year = {2016}, volume = {50}, editor = {Raz, Ran}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.7}, URN = {urn:nbn:de:0030-drops-58513}, doi = {10.4230/LIPIcs.CCC.2016.7}, annote = {Keywords: extractor, derandomization, explicit construction} }

Document

Complete Volume

**Published in:** LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

LIPIcs, Volume 33, CCC'15, Complete Volume

30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@Proceedings{zuckerman:LIPIcs.CCC.2015, title = {{LIPIcs, Volume 33, CCC'15, Complete Volume}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-81-1}, ISSN = {1868-8969}, year = {2015}, volume = {33}, editor = {Zuckerman, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015}, URN = {urn:nbn:de:0030-drops-52111}, doi = {10.4230/LIPIcs.CCC.2015}, annote = {Keywords: Theory of Computation} }

Document

Front Matter

**Published in:** LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

Front Matter, Table of Contents, Preface, Conference Organization

30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. i-xiv, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{zuckerman:LIPIcs.CCC.2015.i, author = {Zuckerman, David}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {i--xiv}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-81-1}, ISSN = {1868-8969}, year = {2015}, volume = {33}, editor = {Zuckerman, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.i}, URN = {urn:nbn:de:0030-drops-50790}, doi = {10.4230/LIPIcs.CCC.2015.i}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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