DROPS

Document

DOI: 10.4230/LIPIcs.ITCS.2025.93

Polynomials, Divided Differences, and Codes

Authors: S. Venkitesh

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract

Multiplicity codes (Kopparty et al., J. ACM 2014) are multivariate polynomial codes where the codewords are described by evaluations of polynomials (with a degree bound) and their derivatives up to some order (the multiplicity parameter), on a suitably chosen affine set of points. While efficient decoding algorithms were known in some special cases of point sets, by a reduction to univariate multiplicity codes, a general algorithm for list decoding up to the distance of the code when the point set is an arbitrary finite grid, was obtained only recently (Bhandari et al., IEEE TIT 2023). This required the characteristic of the field to be zero or larger than the degree bound, which is somewhat necessary since list decoding up to distance with small output list size is not possible when the characteristic is significantly smaller than the degree. In this work, we present an alternative construction based on divided differences of polynomials, that conceptually resembles the classical multiplicity codes but has good list decodability "insensitive to the field characteristic". We obtain a simple algorithm that list decodes this code up to distance for arbitrary finite grids over all finite fields. Our construction can also be interpreted as a folded Reed-Muller code, which may be of independent interest.

Cite as

S. Venkitesh. Polynomials, Divided Differences, and Codes. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 93:1-93:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{venkitesh:LIPIcs.ITCS.2025.93,
  author =	{Venkitesh, S.},
  title =	{{Polynomials, Divided Differences, and Codes}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{93:1--93:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.93},
  URN =		{urn:nbn:de:0030-drops-227216},
  doi =		{10.4230/LIPIcs.ITCS.2025.93},
  annote =	{Keywords: Error-correcting code, polynomial code, Reed-Solomon code, Reed-Muller code, folded Reed-Solomon code, folded Reed-Muller code, multiplicity code, divided difference, q-derivative, polynomial method, list decoding, list decoding capacity, linear algebraic list decoding}
}

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.53

When Do Low-Rate Concatenated Codes Approach The Gilbert-Varshamov Bound?

Authors: Dean Doron, Jonathan Mosheiff, and Mary Wootters

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract

The Gilbert-Varshamov (GV) bound is a classical existential result in coding theory. It implies that a random linear binary code of rate ε² has relative distance at least 1/2 - O(ε) with high probability. However, it is a major challenge to construct explicit codes with similar parameters. One hope to derandomize the Gilbert-Varshamov construction is with code concatenation: We begin with a (hopefully explicit) outer code 𝒞_out over a large alphabet, and concatenate that with a small binary random linear code 𝒞_in. It is known that when we use independent small codes for each coordinate, then the result lies on the GV bound with high probability, but this still uses a lot of randomness. In this paper, we consider the question of whether code concatenation with a single random linear inner code 𝒞_in can lie on the GV bound; and if so what conditions on 𝒞_out are sufficient for this. We show that first, there do exist linear outer codes 𝒞_out that are "good" for concatenation in this sense (in fact, most linear codes codes are good). We also provide two sufficient conditions for 𝒞_out, so that if 𝒞_out satisfies these, 𝒞_out∘𝒞_in will likely lie on the GV bound. We hope that these conditions may inspire future work towards constructing explicit codes 𝒞_out.

Cite as

Dean Doron, Jonathan Mosheiff, and Mary Wootters. When Do Low-Rate Concatenated Codes Approach The Gilbert-Varshamov Bound?. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 53:1-53:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{doron_et_al:LIPIcs.APPROX/RANDOM.2024.53,
  author =	{Doron, Dean and Mosheiff, Jonathan and Wootters, Mary},
  title =	{{When Do Low-Rate Concatenated Codes Approach The Gilbert-Varshamov Bound?}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{53:1--53:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.53},
  URN =		{urn:nbn:de:0030-drops-210467},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.53},
  annote =	{Keywords: Error-correcting codes, Concatenated codes, Derandomization, Gilbert-Varshamov bound}
}

Document

DOI: 10.4230/LIPIcs.CCC.2024.4

Derandomizing Logspace with a Small Shared Hard Drive

Authors: Edward Pyne

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract

We obtain new catalytic algorithms for space-bounded derandomization. In the catalytic computation model introduced by (Buhrman, Cleve, Koucký, Loff, and Speelman STOC 2013), we are given a small worktape, and a larger catalytic tape that has an arbitrary initial configuration. We may edit this tape, but it must be exactly restored to its initial configuration at the completion of the computation. We prove that BPSPACE[S] ⊆ CSPACE[S,S²] where BPSPACE[S] corresponds to randomized space S computation, and CSPACE[S,C] corresponds to catalytic algorithms that use O(S) bits of workspace and O(C) bits of catalytic space. Previously, only BPSPACE[S] ⊆ CSPACE[S,2^O(S)] was known. In fact, we prove a general tradeoff, that for every α ∈ [1,1.5], BPSPACE[S] ⊆ CSPACE[S^α,S^(3-α)]. We do not use the algebraic techniques of prior work on catalytic computation. Instead, we develop an algorithm that branches based on if the catalytic tape is conditionally random, and instantiate this primitive in a recursive framework. Our result gives an alternate proof of the best known time-space tradeoff for BPSPACE[S], due to (Cai, Chakaravarthy, and van Melkebeek, Theory Comput. Sys. 2006). As a final application, we extend our results to solve search problems in CSPACE[S,S²]. As far as we are aware, this constitutes the first study of search problems in the catalytic computing model.

Cite as

Edward Pyne. Derandomizing Logspace with a Small Shared Hard Drive. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{pyne:LIPIcs.CCC.2024.4,
  author =	{Pyne, Edward},
  title =	{{Derandomizing Logspace with a Small Shared Hard Drive}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{4:1--4:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.4},
  URN =		{urn:nbn:de:0030-drops-204006},
  doi =		{10.4230/LIPIcs.CCC.2024.4},
  annote =	{Keywords: Catalytic computation, space-bounded computation, derandomization}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2023.39

Regularization of Low Error PCPs and an Application to MCSP

Authors: Shuichi Hirahara and Dana Moshkovitz

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

Abstract

In a regular PCP the verifier queries each proof symbol in the same number of tests. This number is called the degree of the proof, and it is at least 1/(sq) where s is the soundness error and q is the number of queries. It is incredibly useful to have regularity and reduced degree in PCP. There is an expander-based transformation by Papadimitriou and Yannakakis that transforms any PCP with a constant number of queries and constant soundness error to a regular PCP with constant degree. There are also transformations for low error projection and unique PCPs. Other PCPs are constructed especially to be regular. In this work we show how to regularize and reduce degree of PCPs with a possibly large number of queries and low soundness error. As an application, we prove NP-hardness of an unweighted variant of the collective minimum monotone satisfying assignment problem, which was introduced by Hirahara (FOCS'22) to prove NP-hardness of MCSP^* (the partial function variant of the Minimum Circuit Size Problem) under randomized reductions. We present a simplified proof and sufficient conditions under which MCSP^* is NP-hard under the standard notion of reduction: MCSP^* is NP-hard under deterministic polynomial-time many-one reductions if there exists a function in E that satisfies certain direct sum properties.

Cite as

Shuichi Hirahara and Dana Moshkovitz. Regularization of Low Error PCPs and an Application to MCSP. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 39:1-39:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{hirahara_et_al:LIPIcs.ISAAC.2023.39,
  author =	{Hirahara, Shuichi and Moshkovitz, Dana},
  title =	{{Regularization of Low Error PCPs and an Application to MCSP}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{39:1--39:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.39},
  URN =		{urn:nbn:de:0030-drops-193411},
  doi =		{10.4230/LIPIcs.ISAAC.2023.39},
  annote =	{Keywords: PCP theorem, regularization, Minimum Circuit Size Problem}
}

Document

DOI: 10.4230/LIPIcs.CCC.2023.11

Derandomization with Minimal Memory Footprint

Authors: Dean Doron and Roei Tell

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Abstract

Existing proofs that deduce BPL = 𝐋 from circuit lower bounds convert randomized algorithms into deterministic algorithms with large constant overhead in space. We study space-bounded derandomization with minimal footprint, and ask what is the minimal possible space overhead for derandomization. We show that BPSPACE[S] ⊆ DSPACE[c ⋅ S] for c ≈ 2, assuming space-efficient cryptographic PRGs, and, either: (1) lower bounds against bounded-space algorithms with advice, or: (2) lower bounds against certain uniform compression algorithms. Under additional assumptions regarding the power of catalytic computation, in a new setting of parameters that was not studied before, we are even able to get c ≈ 1. Our results are constructive: Given a candidate hard function (and a candidate cryptographic PRG) we show how to transform the randomized algorithm into an efficient deterministic one. This follows from new PRGs and targeted PRGs for space-bounded algorithms, which we combine with novel space-efficient evaluation methods. A central ingredient in all our constructions is hardness amplification reductions in logspace-uniform TC⁰, that were not known before.

Cite as

Dean Doron and Roei Tell. Derandomization with Minimal Memory Footprint. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{doron_et_al:LIPIcs.CCC.2023.11,
  author =	{Doron, Dean and Tell, Roei},
  title =	{{Derandomization with Minimal Memory Footprint}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{11:1--11:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.11},
  URN =		{urn:nbn:de:0030-drops-182816},
  doi =		{10.4230/LIPIcs.CCC.2023.11},
  annote =	{Keywords: derandomization, space-bounded computation, catalytic space}
}

Document

DOI: 10.4230/LIPIcs.CCC.2022.10

New Near-Linear Time Decodable Codes Closer to the GV Bound

Authors: Guy Blanc and Dean Doron

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract

We construct a family of binary codes of relative distance 1/2-ε and rate ε² ⋅ 2^(-log^α (1/ε)) for α ≈ 1/2 that are decodable, probabilistically, in near-linear time. This improves upon the rate of the state-of-the-art near-linear time decoding near the GV bound due to Jeronimo, Srivastava, and Tulsiani, who gave a randomized decoding of Ta-Shma codes with α ≈ 5/6 [Ta-Shma, 2017; Jeronimo et al., 2021]. Each code in our family can be constructed in probabilistic polynomial time, or deterministic polynomial time given sufficiently good explicit 3-uniform hypergraphs. Our construction is based on a new graph-based bias amplification method. While previous works start with some base code of relative distance 1/2-ε₀ for ε₀ ≫ ε and amplify the distance to 1/2-ε by walking on an expander, or on a carefully tailored product of expanders, we walk over very sparse, highly mixing, hypergraphs. Study of such hypergraphs further offers an avenue toward achieving rate Ω̃(ε²). For our unique- and list-decoding algorithms, we employ the framework developed in [Jeronimo et al., 2021].

Cite as

Guy Blanc and Dean Doron. New Near-Linear Time Decodable Codes Closer to the GV Bound. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 10:1-10:40, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{blanc_et_al:LIPIcs.CCC.2022.10,
  author =	{Blanc, Guy and Doron, Dean},
  title =	{{New Near-Linear Time Decodable Codes Closer to the GV Bound}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{10:1--10:40},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.10},
  URN =		{urn:nbn:de:0030-drops-165726},
  doi =		{10.4230/LIPIcs.CCC.2022.10},
  annote =	{Keywords: Unique decoding, list decoding, the Gilbert-Varshamov bound, small-bias sample spaces, hypergraphs, expander walks}
}

Document

DOI: 10.4230/LIPIcs.CCC.2022.27

Pseudorandomness of Expander Random Walks for Symmetric Functions and Permutation Branching Programs

Authors: Louis Golowich and Salil Vadhan

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract

We study the pseudorandomness of random walks on expander graphs against tests computed by symmetric functions and permutation branching programs. These questions are motivated by applications of expander walks in the coding theory and derandomization literatures. A line of prior work has shown that random walks on expanders with second largest eigenvalue λ fool symmetric functions up to a O(λ) error in total variation distance, but only for the case where the vertices are labeled with symbols from a binary alphabet, and with a suboptimal dependence on the bias of the labeling. We generalize these results to labelings with an arbitrary alphabet, and for the case of binary labelings we achieve an optimal dependence on the labeling bias. We extend our analysis to unify it with and strengthen the expander-walk Chernoff bound. We then show that expander walks fool permutation branching programs up to a O(λ) error in 𝓁₂-distance, and we prove that much stronger bounds hold for programs with a certain structure. We also prove lower bounds to show that our results are tight. To prove our results for symmetric functions, we analyze the Fourier coefficients of the relevant distributions using linear-algebraic techniques. Our analysis for permutation branching programs is likewise linear-algebraic in nature, but also makes use of the recently introduced singular-value approximation notion for matrices (Ahmadinejad et al. 2021).

Cite as

Louis Golowich and Salil Vadhan. Pseudorandomness of Expander Random Walks for Symmetric Functions and Permutation Branching Programs. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 27:1-27:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{golowich_et_al:LIPIcs.CCC.2022.27,
  author =	{Golowich, Louis and Vadhan, Salil},
  title =	{{Pseudorandomness of Expander Random Walks for Symmetric Functions and Permutation Branching Programs}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{27:1--27:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.27},
  URN =		{urn:nbn:de:0030-drops-165893},
  doi =		{10.4230/LIPIcs.CCC.2022.27},
  annote =	{Keywords: Expander graph, Random walk, Pseudorandomness}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2022.55

High-Probability List-Recovery, and Applications to Heavy Hitters

Authors: Dean Doron and Mary Wootters

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract

An error correcting code 𝒞 : Σ^k → Σⁿ is efficiently list-recoverable from input list size 𝓁 if for any sets ℒ₁, …, ℒ_n ⊆ Σ of size at most 𝓁, one can efficiently recover the list ℒ = {x ∈ Σ^k : ∀ j ∈ [n], 𝒞(x)_j ∈ ℒ_j}. While list-recovery has been well-studied in error correcting codes, all known constructions with "efficient" algorithms are not efficient in the parameter 𝓁. In this work, motivated by applications in algorithm design and pseudorandomness, we study list-recovery with the goal of obtaining a good dependence on 𝓁. We make a step towards this goal by obtaining it in the weaker case where we allow a randomized encoding map and a small failure probability, and where the input lists are derived from unions of codewords. As an application of our construction, we give a data structure for the heavy hitters problem in the strict turnstile model that, for some parameter regimes, obtains stronger guarantees than known constructions.

Cite as

Dean Doron and Mary Wootters. High-Probability List-Recovery, and Applications to Heavy Hitters. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 55:1-55:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{doron_et_al:LIPIcs.ICALP.2022.55,
  author =	{Doron, Dean and Wootters, Mary},
  title =	{{High-Probability List-Recovery, and Applications to Heavy Hitters}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{55:1--55:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.55},
  URN =		{urn:nbn:de:0030-drops-163961},
  doi =		{10.4230/LIPIcs.ICALP.2022.55},
  annote =	{Keywords: List recoverable codes, Heavy Hitters, high-dimensional expanders}
}

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.58

Pseudorandom Generators for Read-Once Monotone Branching Programs

Authors: Dean Doron, Raghu Meka, Omer Reingold, Avishay Tal, and Salil Vadhan

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract

Motivated by the derandomization of space-bounded computation, there has been a long line of work on constructing pseudorandom generators (PRGs) against various forms of read-once branching programs (ROBPs), with a goal of improving the O(log² n) seed length of Nisan’s classic construction [Noam Nisan, 1992] to the optimal O(log n). In this work, we construct an explicit PRG with seed length Õ(log n) for constant-width ROBPs that are monotone, meaning that the states at each time step can be ordered so that edges with the same labels never cross each other. Equivalently, for each fixed input, the transition functions are a monotone function of the state. This result is complementary to a line of work that gave PRGs with seed length O(log n) for (ordered) permutation ROBPs of constant width [Braverman et al., 2014; Koucký et al., 2011; De, 2011; Thomas Steinke, 2012], since the monotonicity constraint can be seen as the "opposite" of the permutation constraint. Our PRG also works for monotone ROBPs that can read the input bits in any order, which are strictly more powerful than read-once AC⁰. Our PRG achieves better parameters (in terms of the dependence on the depth of the circuit) than the best previous pseudorandom generator for read-once AC⁰, due to Doron, Hatami, and Hoza [Doron et al., 2019]. Our pseudorandom generator construction follows Ajtai and Wigderson’s approach of iterated pseudorandom restrictions [Ajtai and Wigderson, 1989; Gopalan et al., 2012]. We give a randomness-efficient width-reduction process which proves that the branching program simplifies to an O(log n)-junta after only O(log log n) independent applications of the Forbes-Kelley pseudorandom restrictions [Michael A. Forbes and Zander Kelley, 2018].

Cite as

Dean Doron, Raghu Meka, Omer Reingold, Avishay Tal, and Salil Vadhan. Pseudorandom Generators for Read-Once Monotone Branching Programs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 58:1-58:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{doron_et_al:LIPIcs.APPROX/RANDOM.2021.58,
  author =	{Doron, Dean and Meka, Raghu and Reingold, Omer and Tal, Avishay and Vadhan, Salil},
  title =	{{Pseudorandom Generators for Read-Once Monotone Branching Programs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{58:1--58:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.58},
  URN =		{urn:nbn:de:0030-drops-147513},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.58},
  annote =	{Keywords: Branching programs, pseudorandom generators, constant depth circuits}
}

Document

DOI: 10.4230/LIPIcs.CCC.2021.22

Error Reduction for Weighted PRGs Against Read Once Branching Programs

Authors: Gil Cohen, Dean Doron, Oren Renard, Ori Sberlo, and Amnon Ta-Shma

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Abstract

Weighted pseudorandom generators (WPRGs), introduced by Braverman, Cohen and Garg [Braverman et al., 2020], are a generalization of pseudorandom generators (PRGs) in which arbitrary real weights are considered, rather than a probability mass. Braverman et al. constructed WPRGs against read once branching programs (ROBPs) with near-optimal dependence on the error parameter. Chattopadhyay and Liao [Eshan Chattopadhyay and Jyun-Jie Liao, 2020] somewhat simplified the technically involved BCG construction, also obtaining some improvement in parameters. In this work we devise an error reduction procedure for PRGs against ROBPs. More precisely, our procedure transforms any PRG against length n width w ROBP with error 1/poly(n) having seed length s to a WPRG with seed length s + O(logw/(ε) ⋅ log log1/(ε)). By instantiating our procedure with Nisan’s PRG [Noam Nisan, 1992] we obtain a WPRG with seed length O(log{n} ⋅ log(nw) + logw/(ε) ⋅ log log 1/(ε)). This improves upon [Braverman et al., 2020] and is incomparable with [Eshan Chattopadhyay and Jyun-Jie Liao, 2020]. Our construction is significantly simpler on the technical side and is conceptually cleaner. Another advantage of our construction is its low space complexity O(log{nw})+poly(log log1/(ε)) which is logarithmic in n for interesting values of the error parameter ε. Previous constructions (like [Braverman et al., 2020; Eshan Chattopadhyay and Jyun-Jie Liao, 2020]) specify the seed length but not the space complexity, though it is plausible they can also achieve such (or close) space complexity.

Cite as

Gil Cohen, Dean Doron, Oren Renard, Ori Sberlo, and Amnon Ta-Shma. Error Reduction for Weighted PRGs Against Read Once Branching Programs. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cohen_et_al:LIPIcs.CCC.2021.22,
  author =	{Cohen, Gil and Doron, Dean and Renard, Oren and Sberlo, Ori and Ta-Shma, Amnon},
  title =	{{Error Reduction for Weighted PRGs Against Read Once Branching Programs}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{22:1--22:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.22},
  URN =		{urn:nbn:de:0030-drops-142963},
  doi =		{10.4230/LIPIcs.CCC.2021.22},
  annote =	{Keywords: Pseudorandom generators, Read once branching programs, Space-bounded computation}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2021.7

Pseudorandom Generators for Unbounded-Width Permutation Branching Programs

Authors: William M. Hoza, Edward Pyne, and Salil Vadhan

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract

We prove that the Impagliazzo-Nisan-Wigderson [Impagliazzo et al., 1994] pseudorandom generator (PRG) fools ordered (read-once) permutation branching programs of unbounded width with a seed length of Õ(log d + log n ⋅ log(1/ε)), assuming the program has only one accepting vertex in the final layer. Here, n is the length of the program, d is the degree (equivalently, the alphabet size), and ε is the error of the PRG. In contrast, we show that a randomly chosen generator requires seed length Ω(n log d) to fool such unbounded-width programs. Thus, this is an unusual case where an explicit construction is "better than random." Except when the program’s width w is very small, this is an improvement over prior work. For example, when w = poly(n) and d = 2, the best prior PRG for permutation branching programs was simply Nisan’s PRG [Nisan, 1992], which fools general ordered branching programs with seed length O(log(wn/ε) log n). We prove a seed length lower bound of Ω̃(log d + log n ⋅ log(1/ε)) for fooling these unbounded-width programs, showing that our seed length is near-optimal. In fact, when ε ≤ 1/log n, our seed length is within a constant factor of optimal. Our analysis of the INW generator uses the connection between the PRG and the derandomized square of Rozenman and Vadhan [Rozenman and Vadhan, 2005] and the recent analysis of the latter in terms of unit-circle approximation by Ahmadinejad et al. [Ahmadinejad et al., 2020].

Cite as

William M. Hoza, Edward Pyne, and Salil Vadhan. Pseudorandom Generators for Unbounded-Width Permutation Branching Programs. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hoza_et_al:LIPIcs.ITCS.2021.7,
  author =	{Hoza, William M. and Pyne, Edward and Vadhan, Salil},
  title =	{{Pseudorandom Generators for Unbounded-Width Permutation Branching Programs}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{7:1--7:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.7},
  URN =		{urn:nbn:de:0030-drops-135464},
  doi =		{10.4230/LIPIcs.ITCS.2021.7},
  annote =	{Keywords: Pseudorandom generators, permutation branching programs}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2021.34

Is the Space Complexity of Planted Clique Recovery the Same as That of Detection?

Authors: Jay Mardia

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract

We study the planted clique problem in which a clique of size k is planted in an Erdős-Rényi graph G(n, 1/2), and one is interested in either detecting or recovering this planted clique. This problem is interesting because it is widely believed to show a statistical-computational gap at clique size k = Θ(√n), and has emerged as the prototypical problem with such a gap from which average-case hardness of other statistical problems can be deduced. It also displays a tight computational connection between the detection and recovery variants, unlike other problems of a similar nature. This wide investigation into the computational complexity of the planted clique problem has, however, mostly focused on its time complexity. To begin investigating the robustness of these statistical-computational phenomena to changes in our notion of computational efficiency, we ask- Do the statistical-computational phenomena that make the planted clique an interesting problem also hold when we use "space efficiency" as our notion of computational efficiency? It is relatively easy to show that a positive answer to this question depends on the existence of a O(log n) space algorithm that can recover planted cliques of size k = Ω(√n). Our main result comes very close to designing such an algorithm. We show that for k = Ω(√n), the recovery problem can be solved in O((log^*{n}-log^*{k/(√n}) ⋅ log n) bits of space. 1) If k = ω(√nlog^{(𝓁)}n) for any constant integer 𝓁 > 0, the space usage is O(log n) bits. 2) If k = Θ(√n), the space usage is O(log^* n ⋅ log n) bits. Our result suggests that there does exist an O(log n) space algorithm to recover cliques of size k = Ω(√n), since we come very close to achieving such parameters. This provides evidence that the statistical-computational phenomena that (conjecturally) hold for planted clique time complexity also (conjecturally) hold for space complexity. Due to space limitations, we omit proofs from this manuscript. The entire paper with full proofs can be found on arXiv at https://arxiv.org/abs/2008.12825.

Cite as

Jay Mardia. Is the Space Complexity of Planted Clique Recovery the Same as That of Detection?. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 34:1-34:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{mardia:LIPIcs.ITCS.2021.34,
  author =	{Mardia, Jay},
  title =	{{Is the Space Complexity of Planted Clique Recovery the Same as That of Detection?}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{34:1--34:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.34},
  URN =		{urn:nbn:de:0030-drops-135734},
  doi =		{10.4230/LIPIcs.ITCS.2021.34},
  annote =	{Keywords: Statistical computational gaps, Planted clique, Space complexity, Average case computational complexity}
}

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.7

On Hitting-Set Generators for Polynomials That Vanish Rarely

Authors: Dean Doron, Amnon Ta-Shma, and Roei Tell

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

Abstract

The problem of constructing hitting-set generators for polynomials of low degree is fundamental in complexity theory and has numerous well-known applications. We study the following question, which is a relaxation of this problem: Is it easier to construct a hitting-set generator for polynomials p: 𝔽ⁿ → 𝔽 of degree d if we are guaranteed that the polynomial vanishes on at most an ε > 0 fraction of its inputs? We will specifically be interested in tiny values of ε≪ d/|𝔽|. This question was first considered by Goldreich and Wigderson (STOC 2014), who studied a specific setting geared for a particular application, and another specific setting was later studied by the third author (CCC 2017). In this work our main interest is a systematic study of the relaxed problem, in its general form, and we prove results that significantly improve and extend the two previously-known results. Our contributions are of two types: - Over fields of size 2 ≤ |𝔽| ≤ poly(n), we show that the seed length of any hitting-set generator for polynomials of degree d ≤ n^{.49} that vanish on at most ε = |𝔽|^{-t} of their inputs is at least Ω((d/t)⋅log(n)). - Over 𝔽₂, we show that there exists a (non-explicit) hitting-set generator for polynomials of degree d ≤ n^{.99} that vanish on at most ε = |𝔽|^{-t} of their inputs with seed length O((d-t)⋅log(n)). We also show a polynomial-time computable hitting-set generator with seed length O((d-t)⋅(2^{d-t}+log(n))). In addition, we prove that the problem we study is closely related to the following question: "Does there exist a small set S ⊆ 𝔽ⁿ whose degree-d closure is very large?", where the degree-d closure of S is the variety induced by the set of degree-d polynomials that vanish on S.

Cite as

Dean Doron, Amnon Ta-Shma, and Roei Tell. On Hitting-Set Generators for Polynomials That Vanish Rarely. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{doron_et_al:LIPIcs.APPROX/RANDOM.2020.7,
  author =	{Doron, Dean and Ta-Shma, Amnon and Tell, Roei},
  title =	{{On Hitting-Set Generators for Polynomials That Vanish Rarely}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{7:1--7:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.7},
  URN =		{urn:nbn:de:0030-drops-126109},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.7},
  annote =	{Keywords: Hitting-set generators, Polynomials over finite fields, Quantified derandomization}
}

Document

DOI: 10.4230/LIPIcs.CCC.2020.1

Near-Optimal Erasure List-Decodable Codes

Authors: Avraham Ben-Aroya, Dean Doron, and Amnon Ta-Shma

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Abstract

A code 𝒞 ⊆ {0,1}^n̅ is (s,L) erasure list-decodable if for every word w, after erasing any s symbols of w, the remaining n̅-s symbols have at most L possible completions into a codeword of 𝒞. Non-explicitly, there exist binary ((1-τ)n̅,L) erasure list-decodable codes with rate approaching τ and tiny list-size L = O(log 1/(τ)). Achieving either of these parameters explicitly is a natural open problem (see, e.g., [Guruswami and Indyk, 2002; Guruswami, 2003; Guruswami, 2004]). While partial progress on the problem has been achieved, no prior nontrivial explicit construction achieved rate better than Ω(τ²) or list-size smaller than Ω(1/τ). Furthermore, Guruswami showed no linear code can have list-size smaller than Ω(1/τ) [Guruswami, 2003]. We construct an explicit binary ((1-τ)n̅,L) erasure list-decodable code having rate τ^(1+γ) (for any constant γ > 0 and small τ) and list-size poly(log 1/τ), answering simultaneously both questions, and exhibiting an explicit non-linear code that provably beats the best possible linear code. The binary erasure list-decoding problem is equivalent to the construction of explicit, low-error, strong dispersers outputting one bit with minimal entropy-loss and seed-length. For error ε, no prior explicit construction achieved seed-length better than 2log(1/ε) or entropy-loss smaller than 2log(1/ε), which are the best possible parameters for extractors. We explicitly construct an ε-error one-bit strong disperser with near-optimal seed-length (1+γ)log(1/ε) and entropy-loss O(log log1/ε). The main ingredient in our construction is a new (and almost-optimal) unbalanced two-source extractor. The extractor extracts one bit with constant error from two independent sources, where one source has length n and tiny min-entropy O(log log n) and the other source has length O(log n) and arbitrarily small constant min-entropy rate. When instantiated as a balanced two-source extractor, it improves upon Raz’s extractor [Raz, 2005] in the constant error regime. The construction incorporates recent components and ideas from extractor theory with a delicate and novel analysis needed in order to solve dependency and error issues that prevented previous papers (such as [Li, 2015; Chattopadhyay and Zuckerman, 2019; Cohen, 2016]) from achieving the above results.

Cite as

Avraham Ben-Aroya, Dean Doron, and Amnon Ta-Shma. Near-Optimal Erasure List-Decodable Codes. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 1:1-1:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{benaroya_et_al:LIPIcs.CCC.2020.1,
  author =	{Ben-Aroya, Avraham and Doron, Dean and Ta-Shma, Amnon},
  title =	{{Near-Optimal Erasure List-Decodable Codes}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{1:1--1:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.1},
  URN =		{urn:nbn:de:0030-drops-125531},
  doi =		{10.4230/LIPIcs.CCC.2020.1},
  annote =	{Keywords: Dispersers, Erasure codes, List decoding, Ramsey graphs, Two-source extractors}
}

Document

DOI: 10.4230/LIPIcs.CCC.2020.6

Log-Seed Pseudorandom Generators via Iterated Restrictions

Authors: Dean Doron, Pooya Hatami, and William M. Hoza

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Abstract

There are only a few known general approaches for constructing explicit pseudorandom generators (PRGs). The "iterated restrictions" approach, pioneered by Ajtai and Wigderson [Ajtai and Wigderson, 1989], has provided PRGs with seed length polylog n or even Õ(log n) for several restricted models of computation. Can this approach ever achieve the optimal seed length of O(log n)? In this work, we answer this question in the affirmative. Using the iterated restrictions approach, we construct an explicit PRG for read-once depth-2 AC⁰[⊕] formulas with seed length O(log n) + Õ(log(1/ε)). In particular, we achieve optimal seed length O(log n) with near-optimal error ε = exp(-Ω̃(log n)). Even for constant error, the best prior PRG for this model (which includes read-once CNFs and read-once 𝔽₂-polynomials) has seed length Θ(log n ⋅ (log log n)²) [Chin Ho Lee, 2019]. A key step in the analysis of our PRG is a tail bound for subset-wise symmetric polynomials, a generalization of elementary symmetric polynomials. Like elementary symmetric polynomials, subset-wise symmetric polynomials provide a way to organize the expansion of ∏_{i=1}^m (1 + y_i). Elementary symmetric polynomials simply organize the terms by degree, i.e., they keep track of the number of variables participating in each monomial. Subset-wise symmetric polynomials keep track of more data: for a fixed partition of [m], they keep track of the number of variables from each subset participating in each monomial. Our tail bound extends prior work by Gopalan and Yehudayoff [Gopalan and Yehudayoff, 2014] on elementary symmetric polynomials.

Cite as

Dean Doron, Pooya Hatami, and William M. Hoza. Log-Seed Pseudorandom Generators via Iterated Restrictions. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 6:1-6:36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{doron_et_al:LIPIcs.CCC.2020.6,
  author =	{Doron, Dean and Hatami, Pooya and Hoza, William M.},
  title =	{{Log-Seed Pseudorandom Generators via Iterated Restrictions}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{6:1--6:36},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.6},
  URN =		{urn:nbn:de:0030-drops-125586},
  doi =		{10.4230/LIPIcs.CCC.2020.6},
  annote =	{Keywords: Pseudorandom generators, Pseudorandom restrictions, Read-once depth-2 formulas, Parity gates}
}

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