DROPS

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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.CCC.2024.8

Asymptotically-Good RLCCs with (log n)^(2+o(1)) Queries

Authors: Gil Cohen and Tal Yankovitz

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

Abstract

Recently, Kumar and Mon reached a significant milestone by constructing asymptotically good relaxed locally correctable codes (RLCCs) with poly-logarithmic query complexity. Specifically, they constructed n-bit RLCCs with O(log^{69} n) queries. Their construction relies on a clever reduction to locally testable codes (LTCs), capitalizing on recent breakthrough works in LTCs. As for lower bounds, Gur and Lachish (SICOMP 2021) proved that any asymptotically-good RLCC must make Ω̃(√{log n}) queries. Hence emerges the intriguing question regarding the identity of the least value 1/2 ≤ e ≤ 69 for which asymptotically-good RLCCs with query complexity (log n)^{e+o(1)} exist. In this work, we make substantial progress in narrowing the gap by devising asymptotically-good RLCCs with a query complexity of (log n)^{2+o(1)}. The key insight driving our work lies in recognizing that the strong guarantee of local testability overshoots the requirements for the Kumar-Mon reduction. In particular, we prove that we can replace the LTCs by "vanilla" expander codes which indeed have the necessary property: local testability in the code’s vicinity.

Cite as

Gil Cohen and Tal Yankovitz. Asymptotically-Good RLCCs with (log n)^(2+o(1)) Queries. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cohen_et_al:LIPIcs.CCC.2024.8,
  author =	{Cohen, Gil and Yankovitz, Tal},
  title =	{{Asymptotically-Good RLCCs with (log n)^(2+o(1)) Queries}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{8:1--8:16},
  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.2024.8},
  URN =		{urn:nbn:de:0030-drops-204045},
  doi =		{10.4230/LIPIcs.CCC.2024.8},
  annote =	{Keywords: Relaxed locally decodable codes, Relxaed locally correctable codes, RLCC, RLDC}
}

Document

DOI: 10.4230/LIPIcs.CCC.2024.10

Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs

Authors: Xin Li and Yan Zhong

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

Abstract

Affine extractors give some of the best-known lower bounds for various computational models, such as AC⁰ circuits, parity decision trees, and general Boolean circuits. However, they are not known to give strong lower bounds for read-once branching programs (ROBPs). In a recent work, Gryaznov, Pudlák, and Talebanfard (CCC' 22) introduced a stronger version of affine extractors known as directional affine extractors, together with a generalization of ROBPs where each node can make linear queries, and showed that the former implies strong lower bound for a certain type of the latter known as strongly read-once linear branching programs (SROLBPs). Their main result gives explicit constructions of directional affine extractors for entropy k > 2n/3, which implies average-case complexity 2^{n/3-o(n)} against SROLBPs with exponentially small correlation. A follow-up work by Chattopadhyay and Liao (CCC' 23) improves the hardness to 2^{n-o(n)} at the price of increasing the correlation to polynomially large, via a new connection to sumset extractors introduced by Chattopadhyay and Li (STOC' 16) and explicit constructions of such extractors by Chattopadhyay and Liao (STOC' 22). Both works left open the questions of better constructions of directional affine extractors and improved average-case complexity against SROLBPs in the regime of small correlation. This paper provides a much more in-depth study of directional affine extractors, SROLBPs, and ROBPs. Our main results include: - An explicit construction of directional affine extractors with k = o(n) and exponentially small error, which gives average-case complexity 2^{n-o(n)} against SROLBPs with exponentially small correlation, thus answering the two open questions raised in previous works. - An explicit function in AC⁰ that gives average-case complexity 2^{(1-δ)n} against ROBPs with negligible correlation, for any constant δ > 0. Previously, no such average-case hardness is known, and the best size lower bound for any function in AC⁰ against ROBPs is 2^Ω(n). One of the key ingredients in our constructions is a new linear somewhere condenser for affine sources, which is based on dimension expanders. The condenser also leads to an unconditional improvement of the entropy requirement of explicit affine extractors with negligible error. We further show that the condenser also works for general weak random sources, under the Polynomial Freiman-Ruzsa Theorem in 𝖥₂ⁿ, recently proved by Gowers, Green, Manners, and Tao (arXiv' 23).

Cite as

Xin Li and Yan Zhong. Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{li_et_al:LIPIcs.CCC.2024.10,
  author =	{Li, Xin and Zhong, Yan},
  title =	{{Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{10:1--10:14},
  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.2024.10},
  URN =		{urn:nbn:de:0030-drops-204060},
  doi =		{10.4230/LIPIcs.CCC.2024.10},
  annote =	{Keywords: Randomness Extractors, Affine, Read-once Linear Branching Programs, Low-degree polynomials, AC⁰ circuits}
}

Document

DOI: 10.4230/LIPIcs.CCC.2024.32

BPL ⊆ L-AC¹

Authors: Kuan Cheng and Yichuan Wang

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

Abstract

Whether BPL = 𝖫 (which is conjectured to be equal) or even whether BPL ⊆ NL, is a big open problem in theoretical computer science. It is well known that 𝖫 ⊆ NL ⊆ L-AC¹. In this work we show that BPL ⊆ L-AC¹ also holds. Our proof is based on a new iteration method for boosting precision in approximating matrix powering, which is inspired by the Richardson Iteration method developed in a recent line of work [AmirMahdi Ahmadinejad et al., 2020; Edward Pyne and Salil P. Vadhan, 2021; Gil Cohen et al., 2021; William M. Hoza, 2021; Gil Cohen et al., 2023; Aaron (Louie) Putterman and Edward Pyne, 2023; Lijie Chen et al., 2023]. We also improve the algorithm for approximate counting in low-depth L-AC circuits from an additive error setting to a multiplicative error setting.

Cite as

Kuan Cheng and Yichuan Wang. BPL ⊆ L-AC¹. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cheng_et_al:LIPIcs.CCC.2024.32,
  author =	{Cheng, Kuan and Wang, Yichuan},
  title =	{{BPL ⊆ L-AC¹}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{32:1--32:14},
  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.2024.32},
  URN =		{urn:nbn:de:0030-drops-204282},
  doi =		{10.4230/LIPIcs.CCC.2024.32},
  annote =	{Keywords: Randomized Space Complexity, Circuit Complexity, Derandomization}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2024.74

Linear Relaxed Locally Decodable and Correctable Codes Do Not Need Adaptivity and Two-Sided Error

Authors: Guy Goldberg

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract

Relaxed locally decodable codes (RLDCs) are error-correcting codes in which individual bits of the message can be recovered by querying only a few bits from a noisy codeword. For uncorrupted codewords, and for every bit, the decoder must decode the bit correctly with high probability. However, for a noisy codeword, a relaxed local decoder is allowed to output a "rejection" symbol, indicating that the decoding failed. We study the power of adaptivity and two-sided error for RLDCs. Our main result is that if the underlying code is linear, adaptivity and two-sided error do not give any power to relaxed local decoding. We construct a reduction from adaptive, two-sided error relaxed local decoders to non-adaptive, one-sided error ones. That is, the reduction produces a relaxed local decoder that never errs or rejects if its input is a valid codeword and makes queries based on its internal randomness (and the requested index to decode), independently of the input. The reduction essentially maintains the query complexity, requiring at most one additional query. For any input, the decoder’s error probability increases at most two-fold. Furthermore, assuming the underlying code is in systematic form, where the original message is embedded as the first bits of its encoding, the reduction also conserves both the code itself and its rate and distance properties We base the reduction on our new notion of additive promise problems. A promise problem is additive if the sum of any two YES-instances is a YES-instance and the sum of any NO-instance and a YES-instance is a NO-instance. This novel framework captures both linear RLDCs and property testing (of linear properties), despite their significant differences. We prove that in general, algorithms for any additive promise problem do not gain power from adaptivity or two-sided error, and obtain the result for RLDCs as a special case. The result also holds for relaxed locally correctable codes (RLCCs), where a codeword bit should be recovered. As an application, we improve the best known lower bound for linear adaptive RLDCs. Specifically, we prove that such codes require block length of n ≥ k^{1+Ω(1/q²)}, where k denotes the message length and q denotes the number of queries.

Cite as

Guy Goldberg. Linear Relaxed Locally Decodable and Correctable Codes Do Not Need Adaptivity and Two-Sided Error. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 74:1-74:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{goldberg:LIPIcs.ICALP.2024.74,
  author =	{Goldberg, Guy},
  title =	{{Linear Relaxed Locally Decodable and Correctable Codes Do Not Need Adaptivity and Two-Sided Error}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{74:1--74:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.74},
  URN =		{urn:nbn:de:0030-drops-202174},
  doi =		{10.4230/LIPIcs.ICALP.2024.74},
  annote =	{Keywords: Locally decodable codes, Relaxed locally correctable codes, Relaxed locally decodable codes}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2024.108

Two-Source and Affine Non-Malleable Extractors for Small Entropy

Authors: Xin Li and Yan Zhong

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract

Non-malleable extractors are generalizations and strengthening of standard randomness extractors, that are resilient to adversarial tampering. Such extractors have wide applications in cryptography and have become important cornerstones in recent breakthroughs of explicit constructions of two-source extractors and affine extractors for small entropy. However, explicit constructions of non-malleable extractors appear to be much harder than standard extractors. Indeed, in the well-studied models of two-source and affine non-malleable extractors, the previous best constructions only work for entropy rate > 2/3 and 1-γ for some small constant γ > 0 respectively by Li (FOCS' 23). In this paper, we present explicit constructions of two-source and affine non-malleable extractors that match the state-of-the-art constructions of standard ones for small entropy. Our main results include: - Two-source and affine non-malleable extractors (over 𝖥₂) for sources on n bits with min-entropy k ≥ log^C n and polynomially small error, matching the parameters of standard extractors by Chattopadhyay and Zuckerman (STOC' 16, Annals of Mathematics' 19) and Li (FOCS' 16). - Two-source and affine non-malleable extractors (over 𝖥₂) for sources on n bits with min-entropy k = O(log n) and constant error, matching the parameters of standard extractors by Li (FOCS' 23). Our constructions significantly improve previous results, and the parameters (entropy requirement and error) are the best possible without first improving the constructions of standard extractors. In addition, our improved affine non-malleable extractors give strong lower bounds for a certain kind of read-once linear branching programs, recently introduced by Gryaznov, Pudlák, and Talebanfard (CCC' 22) as a generalization of several well studied computational models. These bounds match the previously best-known average-case hardness results given by Chattopadhyay and Liao (CCC' 23) and Li (FOCS' 23), where the branching program size lower bounds are close to optimal, but the explicit functions we use here are different. Our results also suggest a possible deeper connection between non-malleable extractors and standard ones.

Cite as

Xin Li and Yan Zhong. Two-Source and Affine Non-Malleable Extractors for Small Entropy. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 108:1-108:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{li_et_al:LIPIcs.ICALP.2024.108,
  author =	{Li, Xin and Zhong, Yan},
  title =	{{Two-Source and Affine Non-Malleable Extractors for Small Entropy}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{108:1--108:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.108},
  URN =		{urn:nbn:de:0030-drops-202512},
  doi =		{10.4230/LIPIcs.ICALP.2024.108},
  annote =	{Keywords: Randomness Extractors, Non-malleable, Two-source, Affine}
}

Document

DOI: 10.4230/LIPIcs.CCC.2023.8

Spectral Expanding Expanders

Authors: Gil Cohen and Itay Cohen

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

Abstract

Dinitz, Schapira, and Valadarsky [Dinitz et al., 2017] introduced the intriguing notion of expanding expanders - a family of expander graphs with the property that every two consecutive graphs in the family differ only on a small number of edges. Such a family allows one to add and remove vertices with only few edge updates, making them useful in dynamic settings such as for datacenter network topologies and for the design of distributed algorithms for self-healing expanders. [Dinitz et al., 2017] constructed explicit expanding-expanders based on the Bilu-Linial construction of spectral expanders [Bilu and Linial, 2006]. The construction of expanding expanders, however, ends up being of edge expanders, thus, an open problem raised by [Dinitz et al., 2017] is to construct spectral expanding expanders (SEE). In this work, we resolve this question by constructing SEE with spectral expansion which, like [Bilu and Linial, 2006], is optimal up to a poly-logarithmic factor, and the number of edge updates is optimal up to a constant. We further give a simple proof for the existence of SEE that are close to Ramanujan up to a small additive term. As in [Dinitz et al., 2017], our construction is based on interpolating between a graph and its lift. However, to establish spectral expansion, we carefully weigh the interpolated graphs, dubbed partial lifts, in a way that enables us to conduct a delicate analysis of their spectrum. In particular, at a crucial point in the analysis, we consider the eigenvectors structure of the partial lifts.

Cite as

Gil Cohen and Itay Cohen. Spectral Expanding Expanders. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cohen_et_al:LIPIcs.CCC.2023.8,
  author =	{Cohen, Gil and Cohen, Itay},
  title =	{{Spectral Expanding Expanders}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{8:1--8:19},
  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.2023.8},
  URN =		{urn:nbn:de:0030-drops-182780},
  doi =		{10.4230/LIPIcs.CCC.2023.8},
  annote =	{Keywords: Expanders, Normalized Random Walk, Spectral Analysis}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2022.43

Expander Random Walks: The General Case and Limitations

Authors: Gil Cohen, Dor Minzer, Shir Peleg, Aaron Potechin, and Amnon Ta-Shma

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

Abstract

Cohen, Peri and Ta-Shma [Gil Cohen et al., 2021] considered the following question: Assume the vertices of an expander graph are labelled by ± 1. What "test" functions f : {±1}^t → {±1} can or cannot distinguish t independent samples from those obtained by a random walk? [Gil Cohen et al., 2021] considered only balanced labellings, and proved that for all symmetric functions the distinguishability goes down to zero with the spectral gap λ of the expander G. In addition, [Gil Cohen et al., 2021] show that functions computable by AC⁰ circuits are fooled by expanders with vanishing spectral expansion. We continue the study of this question. We generalize the result to all labelling, not merely balanced ones. We also improve the upper bound on the error of symmetric functions. More importantly, we give a matching lower bound and show a symmetric function with distinguishability going down to zero with λ but not with t. Moreover, we prove a lower bound on the error of functions in AC⁰ in particular, we prove that a random walk on expanders with constant spectral gap does not fool AC⁰.

Cite as

Gil Cohen, Dor Minzer, Shir Peleg, Aaron Potechin, and Amnon Ta-Shma. Expander Random Walks: The General Case and Limitations. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 43:1-43:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cohen_et_al:LIPIcs.ICALP.2022.43,
  author =	{Cohen, Gil and Minzer, Dor and Peleg, Shir and Potechin, Aaron and Ta-Shma, Amnon},
  title =	{{Expander Random Walks: The General Case and Limitations}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{43:1--43:18},
  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.2022.43},
  URN =		{urn:nbn:de:0030-drops-163849},
  doi =		{10.4230/LIPIcs.ICALP.2022.43},
  annote =	{Keywords: Expander Graphs, Random Walks, Lower Bounds}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2022.44

LCC and LDC: Tailor-Made Distance Amplification and a Refined Separation

Authors: Gil Cohen and Tal Yankovitz

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

Abstract

The Alon-Edmonds-Luby distance amplification procedure (FOCS 1995) is an algorithm that transforms a code with vanishing distance to a code with constant distance. AEL was invoked by Kopparty, Meir, Ron-Zewi, and Saraf (J. ACM 2017) for obtaining their state-of-the-art LDC, LCC and LTC. Cohen and Yankovitz (CCC 2021) devised a procedure that can amplify inverse-polynomial distances, exponentially extending the regime of distances that can be amplified by AEL. However, the improved procedure only works for LDC and assuming rate 1-1/(poly log n). In this work we devise a distance amplification procedure for LCC with inverse-polynomial distances even for vanishing rate 1/(poly log log n). For LDC, we obtain a more modest improvement and require rate 1-1/(poly log log n). Thus, the tables have turned and it is now LCC that can be better amplified. Our key idea for accomplishing this, deviating from prior work, is to tailor the distance amplification procedure to the code at hand. Our second result concerns the relation between linear LDC and LCC. We prove the existence of linear LDC that are not LCC, qualitatively extending a separation by Kaufman and Viderman (RANDOM 2010).

Cite as

Gil Cohen and Tal Yankovitz. LCC and LDC: Tailor-Made Distance Amplification and a Refined Separation. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 44:1-44:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cohen_et_al:LIPIcs.ICALP.2022.44,
  author =	{Cohen, Gil and Yankovitz, Tal},
  title =	{{LCC and LDC: Tailor-Made Distance Amplification and a Refined Separation}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{44:1--44:20},
  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.2022.44},
  URN =		{urn:nbn:de:0030-drops-163858},
  doi =		{10.4230/LIPIcs.ICALP.2022.44},
  annote =	{Keywords: Locally Correctable Codes, Locally Decodable Codes, Distance Amplifications}
}

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.54

Candidate Tree Codes via Pascal Determinant Cubes

Authors: Inbar Ben Yaacov, Gil Cohen, and Anand Kumar Narayanan

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

Abstract

Tree codes are combinatorial structures introduced by Schulman [Schulman, 1993] as key ingredients in interactive coding schemes. Asymptotically-good tree codes are long known to exist, yet their explicit construction remains a notoriously hard open problem. Even proposing a plausible construction, without the burden of proof, is difficult and the defining tree code property requires structure that remains elusive. To the best of our knowledge, only one candidate appears in the literature, due to Moore and Schulman [Moore and Schulman, 2014]. We put forth a new candidate for an explicit asymptotically-good tree code. Our construction is an extension of the vanishing rate tree code by Cohen-Haeupler-Schulman [Cohen et al., 2018], and its correctness relies on a conjecture that we introduce on certain Pascal determinants indexed by the points of the Boolean hypercube. Furthermore, using the vanishing distance tree code by Gelles et al. [Gelles et al., 2016] enables us to present a construction that relies on an even weaker assumption. We furnish evidence supporting our conjecture through numerical computation, combinatorial arguments from planar path graphs and based on well-studied heuristics from arithmetic geometry.

Cite as

Inbar Ben Yaacov, Gil Cohen, and Anand Kumar Narayanan. Candidate Tree Codes via Pascal Determinant Cubes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 54:1-54:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{benyaacov_et_al:LIPIcs.APPROX/RANDOM.2021.54,
  author =	{Ben Yaacov, Inbar and Cohen, Gil and Narayanan, Anand Kumar},
  title =	{{Candidate Tree Codes via Pascal Determinant Cubes}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{54:1--54:22},
  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.2021.54},
  URN =		{urn:nbn:de:0030-drops-147474},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.54},
  annote =	{Keywords: Tree codes, Sparse polynomials, Explicit constructions}
}

Document

DOI: 10.4230/LIPIcs.CCC.2021.1

Rate Amplification and Query-Efficient Distance Amplification for Linear LCC and LDC

Authors: Gil Cohen and Tal Yankovitz

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

Abstract

The main contribution of this work is a rate amplification procedure for LCC. Our procedure converts any q-query linear LCC, having rate ρ and, say, constant distance to an asymptotically good LCC with q^poly(1/ρ) queries. Our second contribution is a distance amplification procedure for LDC that converts any linear LDC with distance δ and, say, constant rate to an asymptotically good LDC. The query complexity only suffers a multiplicative overhead that is roughly equal to the query complexity of a length 1/δ asymptotically good LDC. This improves upon the poly(1/δ) overhead obtained by the AEL distance amplification procedure [Alon and Luby, 1996; Alon et al., 1995]. Our work establishes that the construction of asymptotically good LDC and LCC is reduced, with a minor overhead in query complexity, to the problem of constructing a vanishing rate linear LCC and a (rapidly) vanishing distance linear LDC, respectively.

Cite as

Gil Cohen and Tal Yankovitz. Rate Amplification and Query-Efficient Distance Amplification for Linear LCC and LDC. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 1:1-1:57, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cohen_et_al:LIPIcs.CCC.2021.1,
  author =	{Cohen, Gil and Yankovitz, Tal},
  title =	{{Rate Amplification and Query-Efficient Distance Amplification for Linear LCC and LDC}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{1:1--1:57},
  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.2021.1},
  URN =		{urn:nbn:de:0030-drops-142750},
  doi =		{10.4230/LIPIcs.CCC.2021.1},
  annote =	{Keywords: Locally decodable codes, Locally correctable codes}
}

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.2021.48

Pseudobinomiality of the Sticky Random Walk

Authors: Venkatesan Guruswami and Vinayak M. Kumar

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

Abstract

Random walks on expanders are a central and versatile tool in pseudorandomness. If an arbitrary half of the vertices of an expander graph are marked, known Chernoff bounds for expander walks imply that the number M of marked vertices visited in a long n-step random walk strongly concentrates around the expected n/2 value. Surprisingly, it was recently shown that the parity of M also has exponentially small bias. Is there a common unification of these results? What other statistics about M resemble the binomial distribution (the Hamming weight of a random n-bit string)? To gain insight into such questions, we analyze a simpler model called the sticky random walk. This model is a natural stepping stone towards understanding expander random walks, and we also show that it is a necessary step. The sticky random walk starts with a random bit and then each subsequent bit independently equals the previous bit with probability (1+λ)/2. Here λ is the proxy for the expander’s (second largest) eigenvalue. Using Krawtchouk expansion of functions, we derive several probabilistic results about the sticky random walk. We show an asymptotically tight Θ(λ) bound on the total variation distance between the (Hamming weight of the) sticky walk and the binomial distribution. We prove that the correlation between the majority and parity bit of the sticky walk is bounded by O(n^{-1/4}). This lends hope to unifying Chernoff bounds and parity concentration, as well as establishing other interesting statistical properties, of expander random walks.

Cite as

Venkatesan Guruswami and Vinayak M. Kumar. Pseudobinomiality of the Sticky Random Walk. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 48:1-48:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{guruswami_et_al:LIPIcs.ITCS.2021.48,
  author =	{Guruswami, Venkatesan and Kumar, Vinayak M.},
  title =	{{Pseudobinomiality of the Sticky Random Walk}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{48:1--48:19},
  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.2021.48},
  URN =		{urn:nbn:de:0030-drops-135870},
  doi =		{10.4230/LIPIcs.ITCS.2021.48},
  annote =	{Keywords: Expander Graphs, Fourier analysis, Markov Chains, Pseudorandomness, Random Walks}
}

Document

DOI: 10.4230/LIPIcs.CCC.2020.11

Palette-Alternating Tree Codes

Authors: Gil Cohen and Shahar Samocha

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

Abstract

A tree code is an edge-coloring of the complete infinite binary tree such that every two nodes of equal depth have a fraction - bounded away from 0 - of mismatched colors between the corresponding paths to their least common ancestor. Tree codes were introduced in a seminal work by Schulman [Schulman, 1993] and serve as a key ingredient in almost all deterministic interactive coding schemes. The number of colors effects the coding scheme’s rate. It is shown that 4 is precisely the least number of colors for which tree codes exist. Thus, tree-code-based coding schemes cannot achieve rate larger than 1/2. To overcome this barrier, a relaxed notion called palette-alternating tree codes is introduced, in which the number of colors can depend on the layer. We prove the existence of such constructs in which most layers use 2 colors - the bare minimum. The distance-rate tradeoff we obtain matches the Gilbert-Varshamov bound. Based on palette-alternating tree codes, we devise a deterministic interactive coding scheme against adversarial errors that approaches capacity. To analyze our protocol, we prove a structural result on the location of failed communication-rounds induced by the error pattern enforced by the adversary. Our coding scheme is efficient given an explicit palette-alternating tree code and serves as an alternative to the scheme obtained by [R. Gelles et al., 2016].

Cite as

Gil Cohen and Shahar Samocha. Palette-Alternating Tree Codes. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 11:1-11:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cohen_et_al:LIPIcs.CCC.2020.11,
  author =	{Cohen, Gil and Samocha, Shahar},
  title =	{{Palette-Alternating Tree Codes}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{11:1--11:29},
  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.2020.11},
  URN =		{urn:nbn:de:0030-drops-125632},
  doi =		{10.4230/LIPIcs.CCC.2020.11},
  annote =	{Keywords: Tree Codes, Coding Theory, Interactive Coding Scheme}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2020.48

The Computational Cost of Asynchronous Neural Communication

Authors: Yael Hitron, Merav Parter, and Gur Perri

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract

Biological neural computation is inherently asynchronous due to large variations in neuronal spike timing and transmission delays. So-far, most theoretical work on neural networks assumes the synchronous setting where neurons fire simultaneously in discrete rounds. In this work we aim at understanding the barriers of asynchronous neural computation from an algorithmic perspective. We consider an extension of the widely studied model of synchronized spiking neurons [Maass, Neural Networks 97] to the asynchronous setting by taking into account edge and node delays. - Edge Delays: We define an asynchronous model for spiking neurons in which the latency values (i.e., transmission delays) of non self-loop edges vary adversarially over time. This extends the recent work of [Hitron and Parter, ESA'19] in which the latency values are restricted to be fixed over time. Our first contribution is an impossibility result that implies that the assumption that self-loop edges have no delays (as assumed in Hitron and Parter) is indeed necessary. Interestingly, in real biological networks self-loop edges (a.k.a. autapse) are indeed free of delays, and the latter has been noted by neuroscientists to be crucial for network synchronization. To capture the computational challenges in this setting, we first consider the implementation of a single NOT gate. This simple function already captures the fundamental difficulties in the asynchronous setting. Our key technical results are space and time upper and lower bounds for the NOT function, our time bounds are tight. In the spirit of the distributed synchronizers [Awerbuch and Peleg, FOCS'90] and following [Hitron and Parter, ESA'19], we then provide a general synchronizer machinery. Our construction is very modular and it is based on efficient circuit implementation of threshold gates. The complexity of our scheme is measured by the overhead in the number of neurons and the computation time, both are shown to be polynomial in the largest latency value, and the largest incoming degree Δ of the original network. - Node Delays: We introduce the study of asynchronous communication due to variations in the response rates of the neurons in the network. In real brain networks, the round duration varies between different neurons in the network. Our key result is a simulation methodology that allows one to transform the above mentioned synchronized solution under edge delays into a synchronized under node delays while incurring a small overhead w.r.t space and time.

Cite as

Yael Hitron, Merav Parter, and Gur Perri. The Computational Cost of Asynchronous Neural Communication. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 48:1-48:47, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{hitron_et_al:LIPIcs.ITCS.2020.48,
  author =	{Hitron, Yael and Parter, Merav and Perri, Gur},
  title =	{{The Computational Cost of Asynchronous Neural Communication}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{48:1--48:47},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.48},
  URN =		{urn:nbn:de:0030-drops-117330},
  doi =		{10.4230/LIPIcs.ITCS.2020.48},
  annote =	{Keywords: asynchronous communication, asynchronous computation, spiking neurons, synchronizers}
}

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