Document

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

Locally Decodable Codes (LDCs) are error-correcting codes C:Σⁿ → Σ^m, encoding messages in Σⁿ to codewords in Σ^m, with super-fast decoding algorithms. They are important mathematical objects in many areas of theoretical computer science, yet the best constructions so far have codeword length m that is super-polynomial in n, for codes with constant query complexity and constant alphabet size.
In a very surprising result, Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (SICOMP 2006) show how to construct a relaxed version of LDCs (RLDCs) with constant query complexity and almost linear codeword length over the binary alphabet, and used them to obtain significantly-improved constructions of Probabilistically Checkable Proofs.
In this work, we study RLDCs in the standard Hamming-error setting, and introduce their variants in the insertion and deletion (Insdel) error setting. Standard LDCs for Insdel errors were first studied by Ostrovsky and Paskin-Cherniavsky (Information Theoretic Security, 2015), and are further motivated by recent advances in DNA random access bio-technologies.
Our first result is an exponential lower bound on the length of Hamming RLDCs making 2 queries (even adaptively), over the binary alphabet. This answers a question explicitly raised by Gur and Lachish (SICOMP 2021) and is the first exponential lower bound for RLDCs. Combined with the results of Ben-Sasson et al., our result exhibits a "phase-transition"-type behavior on the codeword length for some constant-query complexity. We achieve these lower bounds via a transformation of RLDCs to standard Hamming LDCs, using a careful analysis of restrictions of message bits that fix codeword bits.
We further define two variants of RLDCs in the Insdel-error setting, a weak and a strong version. On the one hand, we construct weak Insdel RLDCs with almost linear codeword length and constant query complexity, matching the parameters of the Hamming variants. On the other hand, we prove exponential lower bounds for strong Insdel RLDCs. These results demonstrate that, while these variants are equivalent in the Hamming setting, they are significantly different in the insdel setting. Our results also prove a strict separation between Hamming RLDCs and Insdel RLDCs.

Alexander R. Block, Jeremiah Blocki, Kuan Cheng, Elena Grigorescu, Xin Li, Yu Zheng, and Minshen Zhu. On Relaxed Locally Decodable Codes for Hamming and Insertion-Deletion Errors. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 14:1-14:25, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{block_et_al:LIPIcs.CCC.2023.14, author = {Block, Alexander R. and Blocki, Jeremiah and Cheng, Kuan and Grigorescu, Elena and Li, Xin and Zheng, Yu and Zhu, Minshen}, title = {{On Relaxed Locally Decodable Codes for Hamming and Insertion-Deletion Errors}}, booktitle = {38th Computational Complexity Conference (CCC 2023)}, pages = {14:1--14:25}, 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.2023.14}, URN = {urn:nbn:de:0030-drops-182847}, doi = {10.4230/LIPIcs.CCC.2023.14}, annote = {Keywords: Relaxed Locally Decodable Codes, Hamming Errors, Insdel Errors, Lower Bounds} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

This work continues the study of linear error correcting codes against adversarial insertion deletion errors (insdel errors). Previously, the work of Cheng, Guruswami, Haeupler, and Li [Kuan Cheng et al., 2021] showed the existence of asymptotically good linear insdel codes that can correct arbitrarily close to 1 fraction of errors over some constant size alphabet, or achieve rate arbitrarily close to 1/2 even over the binary alphabet. As shown in [Kuan Cheng et al., 2021], these bounds are also the best possible. However, known explicit constructions in [Kuan Cheng et al., 2021], and subsequent improved constructions by Con, Shpilka, and Tamo [Con et al., 2022] all fall short of meeting these bounds. Over any constant size alphabet, they can only achieve rate < 1/8 or correct < 1/4 fraction of errors; over the binary alphabet, they can only achieve rate < 1/1216 or correct < 1/54 fraction of errors. Apparently, previous techniques face inherent barriers to achieve rate better than 1/4 or correct more than 1/2 fraction of errors.
In this work we give new constructions of such codes that meet these bounds, namely, asymptotically good linear insdel codes that can correct arbitrarily close to 1 fraction of errors over some constant size alphabet, and binary asymptotically good linear insdel codes that can achieve rate arbitrarily close to 1/2. All our constructions are efficiently encodable and decodable. Our constructions are based on a novel approach of code concatenation, which embeds the index information implicitly into codewords. This significantly differs from previous techniques and may be of independent interest. Finally, we also prove the existence of linear concatenated insdel codes with parameters that match random linear codes, and propose a conjecture about linear insdel codes.

Kuan Cheng, Zhengzhong Jin, Xin Li, Zhide Wei, and Yu Zheng. Linear Insertion Deletion Codes in the High-Noise and High-Rate Regimes. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 41:1-41:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{cheng_et_al:LIPIcs.ICALP.2023.41, author = {Cheng, Kuan and Jin, Zhengzhong and Li, Xin and Wei, Zhide and Zheng, Yu}, title = {{Linear Insertion Deletion Codes in the High-Noise and High-Rate Regimes}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {41:1--41:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.41}, URN = {urn:nbn:de:0030-drops-180931}, doi = {10.4230/LIPIcs.ICALP.2023.41}, annote = {Keywords: Error correcting code, Edit distance, Pseudorandomness, Derandomization} }

Document

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

We study the classical expander codes, introduced by Sipser and Spielman [M. Sipser and D. A. Spielman, 1996]. Given any constants 0 < α, ε < 1/2, and an arbitrary bipartite graph with N vertices on the left, M < N vertices on the right, and left degree D such that any left subset S of size at most α N has at least (1-ε)|S|D neighbors, we show that the corresponding linear code given by parity checks on the right has distance at least roughly {α N}/{2 ε}. This is strictly better than the best known previous result of 2(1-ε) α N [Madhu Sudan, 2000; Viderman, 2013] whenever ε < 1/2, and improves the previous result significantly when ε is small. Furthermore, we show that this distance is tight in general, thus providing a complete characterization of the distance of general expander codes.
Next, we provide several efficient decoding algorithms, which vastly improve previous results in terms of the fraction of errors corrected, whenever ε < 1/4. Finally, we also give a bound on the list-decoding radius of general expander codes, which beats the classical Johnson bound in certain situations (e.g., when the graph is almost regular and the code has a high rate).
Our techniques exploit novel combinatorial properties of bipartite expander graphs. In particular, we establish a new size-expansion tradeoff, which may be of independent interests.

Xue Chen, Kuan Cheng, Xin Li, and Minghui Ouyang. Improved Decoding of Expander Codes. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 43:1-43:3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{chen_et_al:LIPIcs.ITCS.2022.43, author = {Chen, Xue and Cheng, Kuan and Li, Xin and Ouyang, Minghui}, title = {{Improved Decoding of Expander Codes}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {43:1--43:3}, 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.2022.43}, URN = {urn:nbn:de:0030-drops-156394}, doi = {10.4230/LIPIcs.ITCS.2022.43}, annote = {Keywords: Expander Code, Decoding} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

The edit distance (ED) and longest common subsequence (LCS) are two fundamental problems which quantify how similar two strings are to one another. In this paper, we first consider these problems in the asymmetric streaming model introduced by Andoni, Krauthgamer and Onak [Andoni et al., 2010] (FOCS'10) and Saks and Seshadhri [Saks and Seshadhri, 2013] (SODA'13). In this model we have random access to one string and streaming access the other one. Our main contribution is a constant factor approximation algorithm for ED with memory Õ(n^δ) for any constant δ > 0. In addition to this, we present an upper bound of Õ _ε(√n) on the memory needed to approximate ED or LCS within a factor 1±ε. All our algorithms are deterministic and run in polynomial time in a single pass.
We further study small-space approximation algorithms for ED, LCS, and longest increasing sequence (LIS) in the non-streaming setting. Here, we design algorithms that achieve 1 ± ε approximation for all three problems, where ε > 0 can be any constant and even slightly sub-constant. Our algorithms only use poly-logarithmic space while maintaining a polynomial running time. This significantly improves previous results in terms of space complexity, where all known results need to use space at least Ω(√n). Our algorithms make novel use of triangle inequality and carefully designed recursions to save space, which can be of independent interest.

Kuan Cheng, Alireza Farhadi, MohammadTaghi Hajiaghayi, Zhengzhong Jin, Xin Li, Aviad Rubinstein, Saeed Seddighin, and Yu Zheng. Streaming and Small Space Approximation Algorithms for Edit Distance and Longest Common Subsequence. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 54:1-54:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{cheng_et_al:LIPIcs.ICALP.2021.54, author = {Cheng, Kuan and Farhadi, Alireza and Hajiaghayi, MohammadTaghi and Jin, Zhengzhong and Li, Xin and Rubinstein, Aviad and Seddighin, Saeed and Zheng, Yu}, title = {{Streaming and Small Space Approximation Algorithms for Edit Distance and Longest Common Subsequence}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {54:1--54:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.54}, URN = {urn:nbn:de:0030-drops-141236}, doi = {10.4230/LIPIcs.ICALP.2021.54}, annote = {Keywords: Edit Distance, Longest Common Subsequence, Longest Increasing Subsequence, Space Efficient Algorithm, Approximation Algorithm} }

Document

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

A hitting set is a "one-sided" variant of a pseudorandom generator (PRG), naturally suited to derandomizing algorithms that have one-sided error. We study the problem of using a given hitting set to derandomize algorithms that have two-sided error, focusing on space-bounded algorithms. For our first result, we show that if there is a log-space hitting set for polynomial-width read-once branching programs (ROBPs), then not only does 𝐋 = 𝐑𝐋, but 𝐋 = 𝐁𝐏𝐋 as well. This answers a question raised by Hoza and Zuckerman [William M. Hoza and David Zuckerman, 2018].
Next, we consider constant-width ROBPs. We show that if there are log-space hitting sets for constant-width ROBPs, then given black-box access to a constant-width ROBP f, it is possible to deterministically estimate 𝔼[f] to within ± ε in space O(log(n/ε)). Unconditionally, we give a deterministic algorithm for this problem with space complexity O(log² n + log(1/ε)), slightly improving over previous work.
Finally, we investigate the limits of this line of work. Perhaps the strongest reduction along these lines one could hope for would say that for every explicit hitting set, there is an explicit PRG with similar parameters. In the setting of constant-width ROBPs over a large alphabet, we prove that establishing such a strong reduction is at least as difficult as constructing a good PRG outright. Quantitatively, we prove that if the strong reduction holds, then for every constant α > 0, there is an explicit PRG for constant-width ROBPs with seed length O(log^{1 + α} n). Along the way, unconditionally, we construct an improved hitting set for ROBPs over a large alphabet.

Kuan Cheng and William M. Hoza. Hitting Sets Give Two-Sided Derandomization of Small Space. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 10:1-10:25, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{cheng_et_al:LIPIcs.CCC.2020.10, author = {Cheng, Kuan and Hoza, William M.}, title = {{Hitting Sets Give Two-Sided Derandomization of Small Space}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {10:1--10:25}, 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.2020.10}, URN = {urn:nbn:de:0030-drops-125625}, doi = {10.4230/LIPIcs.CCC.2020.10}, annote = {Keywords: hitting sets, derandomization, read-once branching programs} }

Document

Track A: Algorithms, Complexity and Games

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

Document exchange and error correcting codes are two fundamental problems regarding communications. In the first problem, Alice and Bob each holds a string, and the goal is for Alice to send a short sketch to Bob, so that Bob can recover Alice’s string. In the second problem, Alice sends a message with some redundant information to Bob through a channel that can add adversarial errors, and the goal is for Bob to correctly recover the message despite the errors. In both problems, an upper bound is placed on the number of errors between the two strings or that the channel can add, and a major goal is to minimize the size of the sketch or the redundant information. In this paper we focus on deterministic document exchange protocols and binary error correcting codes.
Both problems have been studied extensively. In the case of Hamming errors (i.e., bit substitutions) and bit erasures, we have explicit constructions with asymptotically optimal parameters. However, other error types are still rather poorly understood. In a recent work [Kuan Cheng et al., 2018], the authors constructed explicit deterministic document exchange protocols and binary error correcting codes for edit errors with almost optimal parameters. Unfortunately, the constructions in [Kuan Cheng et al., 2018] do not work for other common errors such as block transpositions.
In this paper, we generalize the constructions in [Kuan Cheng et al., 2018] to handle a much larger class of errors. These include bursts of insertions and deletions, as well as block transpositions. Specifically, we consider document exchange and error correcting codes where the total number of block insertions, block deletions, and block transpositions is at most k <= alpha n/log n for some constant 0<alpha<1. In addition, the total number of bits inserted and deleted by the first two kinds of operations is at most t <= beta n for some constant 0<beta<1, where n is the length of Alice’s string or message. We construct explicit, deterministic document exchange protocols with sketch size O((k log n +t) log^2 n/{k log n + t}) and explicit binary error correcting code with O(k log n log log log n+t) redundant bits. As a comparison, the information-theoretic optimum for both problems is Theta(k log n+t). As far as we know, previously there are no known explicit deterministic document exchange protocols in this case, and the best known binary code needs Omega(n) redundant bits even to correct just one block transposition [L. J. Schulman and D. Zuckerman, 1999].

Kuan Cheng, Zhengzhong Jin, Xin Li, and Ke Wu. Block Edit Errors with Transpositions: Deterministic Document Exchange Protocols and Almost Optimal Binary Codes. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 37:1-37:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{cheng_et_al:LIPIcs.ICALP.2019.37, author = {Cheng, Kuan and Jin, Zhengzhong and Li, Xin and Wu, Ke}, title = {{Block Edit Errors with Transpositions: Deterministic Document Exchange Protocols and Almost Optimal Binary Codes}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {37:1--37:15}, 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.2019.37}, URN = {urn:nbn:de:0030-drops-106137}, doi = {10.4230/LIPIcs.ICALP.2019.37}, annote = {Keywords: Deterministic document exchange, error correcting code, block edit error} }

Document

**Published in:** LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

Randomness extractors, which extract high quality (almost-uniform) random bits from biased random sources, are important objects both in theory and in practice. While there have been significant progress in obtaining near optimal constructions of randomness extractors in various settings, the computational complexity of randomness extractors is still much less studied. In particular, it is not clear whether randomness extractors with good parameters can be computed in several interesting complexity classes that are much weaker than P.
In this paper we study randomness extractors in the following two models of computation: (1) constant-depth circuits (AC^0), and (2) the local computation model. Previous work in these models, such as [Viola, 2005], [Goldreich et al., 2015] and [Bogdanov and Guo, 2013], only achieve constructions with weak parameters. In this work we give explicit constructions of randomness extractors with much better parameters. Our results on AC^0 extractors refute a conjecture in [Goldreich et al., 2015] and answer several open problems there. We also provide a lower bound on the error of extractors in AC^0, which together with the entropy lower bound in [Viola, 2005; Goldreich et al., 2015] almost completely characterizes extractors in this class. Our results on local extractors also significantly improve the seed length in [Bogdanov and Guo, 2013]. As an application, we use our AC^0 extractors to study pseudorandom generators in AC^0, and show that we can construct both cryptographic pseudorandom generators (under reasonable computational assumptions) and unconditional pseudorandom generators for space bounded computation with very good parameters.
Our constructions combine several previous techniques in randomness extractors, as well as introduce new techniques to reduce or preserve the complexity of extractors, which may be of independent interest. These include (1) a general way to reduce the error of strong seeded extractors while preserving the AC^0 property and small locality, and (2) a seeded randomness condenser with small locality.

Kuan Cheng and Xin Li. Randomness Extraction in AC0 and with Small Locality. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 37:1-37:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

Copy BibTex To Clipboard

@InProceedings{cheng_et_al:LIPIcs.APPROX-RANDOM.2018.37, author = {Cheng, Kuan and Li, Xin}, title = {{Randomness Extraction in AC0 and with Small Locality}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {37:1--37:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-085-9}, ISSN = {1868-8969}, year = {2018}, volume = {116}, editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.37}, URN = {urn:nbn:de:0030-drops-94414}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.37}, annote = {Keywords: Randomness Extraction, AC0, Locality, Pseudorandom Generator} }

X

Feedback for Dagstuhl Publishing

Feedback submitted

Please try again later or send an E-mail