20 Search Results for "Levenshtein, Vladimir I."

Document
DOI: 10.4230/LIPIcs.ITCS.2026.23
Decoding Balanced Linear Codes with Preprocessing

Authors: Andrej Bogdanov, Rohit Chatterjee, Yunqi Li, and Prashant Nalini Vasudevan

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
Prange’s information set algorithm is a well-known decoding algorithm for linear codes. It decodes corrupted codewords of most 𝔽₂-linear codes C of message length n up to relative error rate O(log n / n) in poly(n) time. We show that the error rate can be improved to O((log n)² / n), provided: (1) the decoder has access to a polynomial-length advice string that depends on C only, and (2) C is n^{-Ω(1)}-balanced. As a consequence we improve the error tolerance in decoding random linear codes if inefficient preprocessing of the code is allowed. This reveals potential vulnerabilities in cryptographic applications of Learning Noisy Parities with low noise rate. Our main technical result is that the Hamming weight of Hw, where the rows of H are a random sample of short dual codewords, measures the proximity of a received word w to the code in the regime of interest. Given such H as advice, our algorithm corrects errors by locally minimizing this measure. We show that for most codes, the error rate tolerated by our decoder is asymptotically optimal among all algorithms whose decision is based on thresholding Hw for an arbitrary polynomial-size advice matrix H.
Cite as

Andrej Bogdanov, Rohit Chatterjee, Yunqi Li, and Prashant Nalini Vasudevan. Decoding Balanced Linear Codes with Preprocessing. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 23:1-23:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bogdanov_et_al:LIPIcs.ITCS.2026.23,
  author =	{Bogdanov, Andrej and Chatterjee, Rohit and Li, Yunqi and Vasudevan, Prashant Nalini},
  title =	{{Decoding Balanced Linear Codes with Preprocessing}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{23:1--23:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  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.ITCS.2026.23},
  URN =		{urn:nbn:de:0030-drops-253107},
  doi =		{10.4230/LIPIcs.ITCS.2026.23},
  annote =	{Keywords: Linear codes, nearest codeword problem, learning parity with noise}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.44
Higher-Order Delsarte Dual LPs: Lifting, Constructions and Completeness

Authors: Leonardo Nagami Coregliano, Fernando Granha Jeronimo, Chris Jones, Nati Linial, and Elyassaf Loyfer

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
A central and longstanding open problem in coding theory is the rate-versus-distance trade-off for binary error-correcting codes. In a seminal work, Delsarte introduced a family of linear programs establishing relaxations on the size of optimum codes. To date, the state-of-the-art upper bounds for binary codes come from dual feasible solutions to these LPs. Still, these bounds are exponentially far from the best-known existential constructions. Recently, hierarchies of linear programs extending and strengthening Delsarte’s original LPs were introduced for linear codes, which we refer to as higher-order Delsarte LPs. These new hierarchies were shown to provably converge to the actual value of optimum codes, namely, they are complete hierarchies. Therefore, understanding them and their dual formulations becomes a valuable line of investigation. Nonetheless, their higher-order structure poses challenges. In fact, analysis of all known convex programming hierarchies strengthening Delsarte’s original LPs has turned out to be exceedingly difficult and essentially nothing is known, stalling progress in the area since the 1970s. Our main result is an analysis of the higher-order Delsarte LPs via their dual formulation. Although quantitatively, our current analysis only matches the best-known upper bounds, it shows, for the first time, how to tame the complexity of analyzing a hierarchy strengthening Delsarte’s original LPs. In doing so, we reach a better understanding of the structure of the hierarchy, which may serve as the foundation for further quantitative improvements. We provide two additional structural results for this hierarchy. First, we show how to explicitly lift any feasible dual solution from level k to a (suitable) larger level 𝓁 while retaining the objective value. Second, we give a novel proof of completeness using the dual formulation.
Cite as

Leonardo Nagami Coregliano, Fernando Granha Jeronimo, Chris Jones, Nati Linial, and Elyassaf Loyfer. Higher-Order Delsarte Dual LPs: Lifting, Constructions and Completeness. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 44:1-44:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{coregliano_et_al:LIPIcs.ITCS.2026.44,
  author =	{Coregliano, Leonardo Nagami and Jeronimo, Fernando Granha and Jones, Chris and Linial, Nati and Loyfer, Elyassaf},
  title =	{{Higher-Order Delsarte Dual LPs: Lifting, Constructions and Completeness}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{44:1--44:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  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.ITCS.2026.44},
  URN =		{urn:nbn:de:0030-drops-253315},
  doi =		{10.4230/LIPIcs.ITCS.2026.44},
  annote =	{Keywords: Coding theory, code bounds, convex optimization, linear progamming hierarchy}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.30
New Bounds for Circular Trace Reconstruction

Authors: Arnav Burudgunte, Paul Valiant, and Hongao Wang

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
The "trace reconstruction" problem asks, given an unknown binary string x and a channel that repeatedly returns "traces" of x with each bit randomly deleted with some probability p, how many traces are needed to recover x? There is an exponential gap between the best known upper and lower bounds for this problem. Many variants of the model have been introduced in hopes of motivating or revealing new approaches to narrow this gap. We study the variant of circular trace reconstruction introduced by Narayanan and Ren (ITCS 2021), in which traces undergo a random cyclic shift in addition to random deletions. We show an improved lower bound of Ω̃(n⁵) for circular trace reconstruction. This contrasts with the (previously) best known lower bounds of Ω̃(n³) in the circular case and Ω̃(n^{3/2}) in the linear case. Our bound shows the indistinguishability of traces from two sparse strings x,y that each have a constant number of nonzeros. Can this technique be extended significantly? How hard is it to reconstruct a sparse string x under a cyclic deletion channel? We resolve these questions by showing, using Fourier techniques, that Õ(n⁶) traces suffice for reconstructing any constant-sparse string in a circular deletion channel, in contrast to the best known upper bound of exp(Õ(n^{1/3})) for general strings in the circular deletion channel. This shows that new algorithms or new lower bounds must focus on non-constant-sparse strings.
Cite as

Arnav Burudgunte, Paul Valiant, and Hongao Wang. New Bounds for Circular Trace Reconstruction. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 30:1-30:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{burudgunte_et_al:LIPIcs.ITCS.2026.30,
  author =	{Burudgunte, Arnav and Valiant, Paul and Wang, Hongao},
  title =	{{New Bounds for Circular Trace Reconstruction}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{30:1--30:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  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.ITCS.2026.30},
  URN =		{urn:nbn:de:0030-drops-253176},
  doi =		{10.4230/LIPIcs.ITCS.2026.30},
  annote =	{Keywords: Trace reconstruction, algorithmic statistics, Fourier analysis}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.45
Bounded Weighted Edit Distance: Dynamic Algorithms and Matching Lower Bounds

Authors: Itai Boneh, Egor Gorbachev, and Tomasz Kociumaka

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
The edit distance ed(X,Y) of two strings X,Y ∈ Σ^* is the minimum number of character edits (insertions, deletions, and substitutions) needed to transform X into Y. Its weighted counterpart ed^w(X,Y) minimizes the total cost of edits, where the costs of individual edits, depending on the edit type and the characters involved, are specified using a function w, normalized so that each edit costs at least one. The textbook dynamic-programming procedure, given strings X,Y ∈ Σ^{≤ n} and oracle access to w, computes ed^w(X,Y) in 𝒪(n²) time. Nevertheless, one can achieve better running times if the computed distance, denoted k, is small: 𝒪(n+k²) for unit weights [Landau and Vishkin; JCSS'88] and Õ(n+√{nk³}) for arbitrary weights [Cassis, Kociumaka, Wellnitz; FOCS'23]. In this paper, we study the dynamic version of the weighted edit distance problem, where the goal is to maintain ed^w(X,Y) for strings X,Y ∈ Σ^{≤ n} that change over time, with each update specified as an edit in X or Y. Very recently, Gorbachev and Kociumaka [STOC'25] showed that the unweighted distance ed(X,Y) can be maintained in Õ(k) time per update after Õ(n+k²)-time preprocessing; here, k denotes the current value of ed(X,Y). Their algorithm generalizes to small integer weights, but the underlying approach is incompatible with large weights. Our main result is a dynamic algorithm that maintains ed^w(X,Y) in Õ(k^{3-γ}) time per update after Õ(nk^γ)-time preprocessing. Here, γ ∈ [0,1] is a real trade-off parameter and k ≥ 1 is an integer threshold fixed at preprocessing time, with ∞ returned whenever ed^w(X,Y) > k. We complement our algorithm with conditional lower bounds showing fine-grained optimality of our trade-off for γ ∈ [0.5,1) and justifying our choice to fix k. We also generalize our solution to a much more robust setting while preserving the fine-grained optimal trade-off. Our full algorithm maintains X ∈ Σ^{≤ n} subject not only to character edits but also substring deletions and copy-pastes, each supported in Õ(k²) time. Instead of dynamically maintaining Y, it answers queries that, given any string Y specified through a sequence of 𝒪(k) arbitrary edits transforming X into Y, in Õ(k^{3-γ}) time compute ed^w(X,Y) or report that ed^w(X,Y) > k.
Cite as

Itai Boneh, Egor Gorbachev, and Tomasz Kociumaka. Bounded Weighted Edit Distance: Dynamic Algorithms and Matching Lower Bounds. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 45:1-45:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{boneh_et_al:LIPIcs.ESA.2025.45,
  author =	{Boneh, Itai and Gorbachev, Egor and Kociumaka, Tomasz},
  title =	{{Bounded Weighted Edit Distance: Dynamic Algorithms and Matching Lower Bounds}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{45:1--45:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.45},
  URN =		{urn:nbn:de:0030-drops-245139},
  doi =		{10.4230/LIPIcs.ESA.2025.45},
  annote =	{Keywords: Edit distance, dynamic algorithms, conditional lower bounds}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.77
Minimization of Deterministic Finite Automata Modulo the Edit Distance

Authors: Jakub Michaliszyn and Jan Otop

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
We propose a novel approach to minimization of deterministic finite automata (DFA), in which the DFA is further minimized at the expense of relaxing equality of languages to merely a similarity. As the notion of similarity of languages, we consider the edit distance between languages ℒ, ℒ', i.e., the minimal number of edits necessary to transform any word from ℒ to some word from ℒ' and vice versa. In this paper we address two problems: minimization up to a predetermined edit distance given in the input, and minimization up to a bounded edit distance, in which there has to be an upper bound on the number of edits, but it is not specified. We show the first problem to be PSpace {}-complete and that the second problem is in Σ₂^p, and both NP-hard and coNP-hard. We show that if we limit how many strongly connected components can be visited by a single run (i.e., bounded SCC-depth), the problem becomes NP-complete. We also establish maximal subclasses of DFA over which minimization up to a bounded edit distance can be performed in polynomial time. Additionally, we provide a succinct overview of alternative metrics for assessing language similarity.
Cite as

Jakub Michaliszyn and Jan Otop. Minimization of Deterministic Finite Automata Modulo the Edit Distance. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 77:1-77:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{michaliszyn_et_al:LIPIcs.MFCS.2025.77,
  author =	{Michaliszyn, Jakub and Otop, Jan},
  title =	{{Minimization of Deterministic Finite Automata Modulo the Edit Distance}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{77:1--77:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.77},
  URN =		{urn:nbn:de:0030-drops-241843},
  doi =		{10.4230/LIPIcs.MFCS.2025.77},
  annote =	{Keywords: automata theory, automata minimization, edit distance}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.7
Sparser Abelian High Dimensional Expanders

Authors: Yotam Dikstein, Siqi Liu, and Avi Wigderson

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
The focus of this paper is the development of new elementary techniques for the construction and analysis of high dimensional expanders. Specifically, we present two new explicit constructions of Cayley high dimensional expanders (HDXs) over the abelian group 𝔽₂ⁿ. Our expansion proofs use only linear algebra and combinatorial arguments. The first construction gives local spectral HDXs of any constant dimension and subpolynomial degree exp(n^ε) for every ε > 0, improving on a construction by Golowich [Golowich, 2023] which achieves ε = 1/2. [Golowich, 2023] derives these HDXs by sparsifying the complete Grassmann poset of subspaces. The novelty in our construction is the ability to sparsify any expanding Grassmann posets, leading to iterated sparsification and much smaller degrees. The sparse Grassmannian (which is of independent interest in the theory of HDXs) serves as the generating set of the Cayley graph. Our second construction gives a 2-dimensional HDX of any polynomial degree exp(ε n) for any constant ε > 0, which is simultaneously a spectral expander and a coboundary expander. To the best of our knowledge, this is the first such non-trivial construction. We name it the Johnson complex, as it is derived from the classical Johnson scheme, whose vertices serve as the generating set of this Cayley graph. This construction may be viewed as a derandomization of the recent random geometric complexes of [Liu et al., 2023]. Establishing coboundary expansion through Gromov’s "cone method" and the associated isoperimetric inequalities is the most intricate aspect of this construction. While these two constructions are quite different, we show that they both share a common structure, resembling the intersection patterns of vectors in the Hadamard code. We propose a general framework of such "Hadamard-like" constructions in the hope that it will yield new HDXs.
Cite as

Yotam Dikstein, Siqi Liu, and Avi Wigderson. Sparser Abelian High Dimensional Expanders. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 7:1-7:98, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dikstein_et_al:LIPIcs.CCC.2025.7,
  author =	{Dikstein, Yotam and Liu, Siqi and Wigderson, Avi},
  title =	{{Sparser Abelian High Dimensional Expanders}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{7:1--7:98},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.7},
  URN =		{urn:nbn:de:0030-drops-237013},
  doi =		{10.4230/LIPIcs.CCC.2025.7},
  annote =	{Keywords: Local spectral expander, coboundary expander, Grassmannian expander}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.3
Near-Optimal Trace Reconstruction for Mildly Separated Strings

Authors: Anders Aamand, Allen Liu, and Shyam Narayanan

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
In the trace reconstruction problem our goal is to learn an unknown string x ∈ {0,1}ⁿ given independent traces of x. A trace is obtained by independently deleting each bit of x with some probability δ and concatenating the remaining bits. It is a major open question whether the trace reconstruction problem can be solved with a polynomial number of traces when the deletion probability δ is constant. The best known upper bound and lower bounds are respectively exp(Õ(n^{1/5})) [Zachary Chase, 2021a] and ̃ Ω(n^{3/2}) [Zachary Chase, 2021b]. Our main result is that if the string x is mildly separated, meaning that the number of zeros between any two ones in x is at least polylog n, and if δ is a sufficiently small constant, then the trace reconstruction problem can be solved with O(n log n) traces and in polynomial time.
Cite as

Anders Aamand, Allen Liu, and Shyam Narayanan. Near-Optimal Trace Reconstruction for Mildly Separated Strings. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 3:1-3:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{aamand_et_al:LIPIcs.ICALP.2025.3,
  author =	{Aamand, Anders and Liu, Allen and Narayanan, Shyam},
  title =	{{Near-Optimal Trace Reconstruction for Mildly Separated Strings}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{3:1--3:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l 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.2025.3},
  URN =		{urn:nbn:de:0030-drops-233801},
  doi =		{10.4230/LIPIcs.ICALP.2025.3},
  annote =	{Keywords: Trace Reconstruction, deletion channel, sample complexity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.60
Random Reed-Solomon Codes Achieve the Half-Singleton Bound for Insertions and Deletions over Linear-Sized Alphabets

Authors: Roni Con, Zeyu Guo, Ray Li, and Zihan Zhang

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
In this paper, we prove that with high probability, random Reed-Solomon codes approach the half-Singleton bound - the optimal rate versus error tradeoff for linear insdel codes - with linear-sized alphabets. More precisely, we prove that, for any ε > 0 and positive integers n and k, with high probability, random Reed-Solomon codes of length n and dimension k can correct (1-ε)n-2k+1 adversarial insdel errors over alphabets of size n+2^{poly(1/ε)}k. This significantly improves upon the alphabet size demonstrated in the work of Con, Shpilka, and Tamo (IEEE TIT, 2023), who showed the existence of Reed-Solomon codes with exponential alphabet size Õ(binom(n,2k-1)²) precisely achieving the half-Singleton bound. Our methods are inspired by recent works on list-decoding Reed-Solomon codes. Brakensiek-Gopi-Makam (STOC 2023) showed that random Reed-Solomon codes are list-decodable up to capacity with exponential-sized alphabets, and Guo-Zhang (FOCS 2023) and Alrabiah-Guruswami-Li (STOC 2024) improved the alphabet-size to linear. We achieve a similar alphabet-size reduction by similarly establishing strong bounds on the probability that certain random rectangular matrices are full rank. To accomplish this in our insdel context, our proof combines the random matrix techniques from list-decoding with structural properties of Longest Common Subsequences.
Cite as

Roni Con, Zeyu Guo, Ray Li, and Zihan Zhang. Random Reed-Solomon Codes Achieve the Half-Singleton Bound for Insertions and Deletions over Linear-Sized Alphabets. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 60:1-60:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{con_et_al:LIPIcs.ICALP.2025.60,
  author =	{Con, Roni and Guo, Zeyu and Li, Ray and Zhang, Zihan},
  title =	{{Random Reed-Solomon Codes Achieve the Half-Singleton Bound for Insertions and Deletions over Linear-Sized Alphabets}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{60:1--60:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l 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.2025.60},
  URN =		{urn:nbn:de:0030-drops-234372},
  doi =		{10.4230/LIPIcs.ICALP.2025.60},
  annote =	{Keywords: coding theory, error-correcting codes, Reed-Solomon codes, insdel, insertion-deletion errors, half-Singleton bound}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.82
Tight Bounds on List-Decodable and List-Recoverable Zero-Rate Codes

Authors: Nicolas Resch, Chen Yuan, and Yihan Zhang

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

Abstract
In this work, we consider the list-decodability and list-recoverability of codes in the zero-rate regime. Briefly, a code 𝒞 ⊆ [q]ⁿ is (p,𝓁,L)-list-recoverable if for all tuples of input lists (Y₁,… ,Y_n) with each Y_i ⊆ [q] and |Y_i| = 𝓁, the number of codewords c ∈ 𝒞 such that c_i ∉ Y_i for at most pn choices of i ∈ [n] is less than L; list-decoding is the special case of 𝓁 = 1. In recent work by Resch, Yuan and Zhang (ICALP 2023) the zero-rate threshold for list-recovery was determined for all parameters: that is, the work explicitly computes p_*: = p_*(q,𝓁,L) with the property that for all ε > 0 (a) there exist positive-rate (p_*-ε,𝓁,L)-list-recoverable codes, and (b) any (p_*+ε,𝓁,L)-list-recoverable code has rate 0. In fact, in the latter case the code has constant size, independent on n. However, the constant size in their work is quite large in 1/ε, at least |𝒞| ≥ (1/(ε))^O(q^L). Our contribution in this work is to show that for all choices of q,𝓁 and L with q ≥ 3, any (p_*+ε,𝓁,L)-list-recoverable code must have size O_{q,𝓁,L}(1/ε), and furthermore this upper bound is complemented by a matching lower bound Ω_{q,𝓁,L}(1/ε). This greatly generalizes work by Alon, Bukh and Polyanskiy (IEEE Trans. Inf. Theory 2018) which focused only on the case of binary alphabet (and thus necessarily only list-decoding). We remark that we can in fact recover the same result for q = 2 and even L, as obtained by Alon, Bukh and Polyanskiy: we thus strictly generalize their work. Our main technical contribution is to (a) properly define a linear programming relaxation of the list-recovery condition over large alphabets; and (b) to demonstrate that a certain function defined on a q-ary probability simplex is maximized by the uniform distribution. This represents the core challenge in generalizing to larger q (as a binary simplex can be naturally identified with a one-dimensional interval). We can subsequently re-utilize certain Schur convexity and convexity properties established for a related function by Resch, Yuan and Zhang along with ideas of Alon, Bukh and Polyanskiy.
Cite as

Nicolas Resch, Chen Yuan, and Yihan Zhang. Tight Bounds on List-Decodable and List-Recoverable Zero-Rate Codes. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 82:1-82:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{resch_et_al:LIPIcs.ITCS.2025.82,
  author =	{Resch, Nicolas and Yuan, Chen and Zhang, Yihan},
  title =	{{Tight Bounds on List-Decodable and List-Recoverable Zero-Rate Codes}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{82:1--82:21},
  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.2025.82},
  URN =		{urn:nbn:de:0030-drops-227103},
  doi =		{10.4230/LIPIcs.ITCS.2025.82},
  annote =	{Keywords: List Decoding, List Recovery, Zero Rate}
}
Document
DOI: 10.4230/DagSemProc.06201.3
Local Minimax Learning of Approximately Polynomial Functions

Authors: Lee Jones and Konstantin Rybnikov

Published in: Dagstuhl Seminar Proceedings, Volume 6201, Combinatorial and Algorithmic Foundations of Pattern and Association Discovery (2006)

Abstract
Suppose we have a number of noisy measurements of an unknown real-valued function $f$ near point of interest $mathbf{x}_0 in mathbb{R}^d$. Suppose also that nothing can be assumed about the noise distribution, except for zero mean and bounded covariance matrix. We want to estimate $f$ at $mathbf{x=x}_0$ using a general linear parametric family $f(mathbf{x};mathbf{a}) = a_0 h_0 (mathbf{x}) ++ a_q h_q (mathbf{x})$, where $mathbf{a} in mathbb{R}^q$ and $h_i$'s are bounded functions on a neighborhood $B$ of $mathbf{x}_0$ which contains all points of measurement. Typically, $B$ is a Euclidean ball or cube in $mathbb{R}^d$ (more generally, a ball in an $l_p$-norm). In the case when the $h_i$'s are polynomial functions in $x_1,ldots,x_d$ the model is called locally-polynomial. In particular, if the $h_i$'s form a basis of the linear space of polynomials of degree at most two, the model is called locally-quadratic (if the degree is at most three, the model is locally-cubic, etc.). Often, there is information, which is called context, about the function $f$ (restricted to $B$ ) available, such as that it takes values in a known interval, or that it satisfies a Lipschitz condition. The theory of local minimax estimation with context for locally-polynomial models and approximately locally polynomial models has been recently initiated by Jones. In the case of local linearity and a bound on the change of $f$ on $B$, where $B$ is a ball, the solution for squared error loss is in the form of ridge regression, where the ridge parameter is identified; hence, minimax justification for ridge regression is given together with explicit best error bounds. The analysis of polynomial models of degree above 1 leads to interesting and difficult questions in real algebraic geometry and non-linear optimization. We show that in the case when $f$ is a probability function, the optimal (in the minimax sense) estimator is effectively computable (with any given precision), thanks to Tarski's elimination principle.
Cite as

Lee Jones and Konstantin Rybnikov. Local Minimax Learning of Approximately Polynomial Functions. In Combinatorial and Algorithmic Foundations of Pattern and Association Discovery. Dagstuhl Seminar Proceedings, Volume 6201, pp. 1-12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{jones_et_al:DagSemProc.06201.3,
  author =	{Jones, Lee and Rybnikov, Konstantin},
  title =	{{Local Minimax Learning of Approximately Polynomial Functions}},
  booktitle =	{Combinatorial and Algorithmic Foundations of Pattern and Association Discovery},
  pages =	{1--12},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{6201},
  editor =	{Rudolf Ahlswede and Alberto Apostolico and Vladimir I. Levenshtein},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06201.3},
  URN =		{urn:nbn:de:0030-drops-8912},
  doi =		{10.4230/DagSemProc.06201.3},
  annote =	{Keywords: Local learning, statistical learning, estimator, minimax, convex optimization, quantifier elimination, semialgebraic, ridge regression, polynomial}
}
Document
DOI: 10.4230/DagSemProc.06201.7
Solving Classical String Problems an Compressed Texts

Authors: Yury Lifshits

Published in: Dagstuhl Seminar Proceedings, Volume 6201, Combinatorial and Algorithmic Foundations of Pattern and Association Discovery (2006)

Abstract
How to solve string problems, if instead of input string we get only program generating it? Is it possible to solve problems faster than just "generate text + apply classical algorithm"? In this paper we consider strings generated by straight-line programs (SLP). These are programs using only assignment operator. We show new algorithms for equivalence, pattern matching, finding periods and covers, computing fingerprint table on SLP-generated strings. From the other hand, computing the Hamming distance is NP-hard. Main corollary is an $O(n2*m)$ algorithm for pattern matching in LZ-compressed texts.
Cite as

Yury Lifshits. Solving Classical String Problems an Compressed Texts. In Combinatorial and Algorithmic Foundations of Pattern and Association Discovery. Dagstuhl Seminar Proceedings, Volume 6201, pp. 1-10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{lifshits:DagSemProc.06201.7,
  author =	{Lifshits, Yury},
  title =	{{Solving Classical String Problems an Compressed Texts}},
  booktitle =	{Combinatorial and Algorithmic Foundations of Pattern and Association Discovery},
  pages =	{1--10},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6201},
  editor =	{Rudolf Ahlswede and Alberto Apostolico and Vladimir I. Levenshtein},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06201.7},
  URN =		{urn:nbn:de:0030-drops-7984},
  doi =		{10.4230/DagSemProc.06201.7},
  annote =	{Keywords: Pattern matching, Compressed text}
}
Document
DOI: 10.4230/DagSemProc.06201.1
06201 Abstracts Collection – Combinatorial and Algorithmic Foundations of Pattern and Association Discovery

Authors: Rudolf Ahlswede, Alberto Apostolico, and Vladimir I. Levenshtein

Published in: Dagstuhl Seminar Proceedings, Volume 6201, Combinatorial and Algorithmic Foundations of Pattern and Association Discovery (2006)

Abstract
From 15.05.06 to 20.05.06, the Dagstuhl Seminar 06201 ``Combinatorial and Algorithmic Foundations of Pattern and Association Discovery'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.
Cite as

Rudolf Ahlswede, Alberto Apostolico, and Vladimir I. Levenshtein. 06201 Abstracts Collection – Combinatorial and Algorithmic Foundations of Pattern and Association Discovery. In Combinatorial and Algorithmic Foundations of Pattern and Association Discovery. Dagstuhl Seminar Proceedings, Volume 6201, pp. 1-15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{ahlswede_et_al:DagSemProc.06201.1,
  author =	{Ahlswede, Rudolf and Apostolico, Alberto and Levenshtein, Vladimir I.},
  title =	{{06201 Abstracts Collection – Combinatorial and Algorithmic Foundations of Pattern and Association Discovery}},
  booktitle =	{Combinatorial and Algorithmic Foundations of Pattern and Association Discovery},
  pages =	{1--15},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6201},
  editor =	{Rudolf Ahlswede and Alberto Apostolico and Vladimir I. Levenshtein},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06201.1},
  URN =		{urn:nbn:de:0030-drops-7873},
  doi =		{10.4230/DagSemProc.06201.1},
  annote =	{Keywords: Data compression, pattern matching, pattern discovery, search, sorting, molecular biology, reconstruction, genome rearrangements}
}
Document
DOI: 10.4230/DagSemProc.06201.2
06201 Executive Summary – Combinatorial and Algorithmic Foundations of Pattern and Association Discovery

Authors: Rudolf Ahlswede, Alberto Apostolico, and Vladimir I. Levenshtein

Published in: Dagstuhl Seminar Proceedings, Volume 6201, Combinatorial and Algorithmic Foundations of Pattern and Association Discovery (2006)

Abstract
The goals of this seminar have been (1) to identify and match recently developed methods to specific tasks and data sets in a core of application areas; next, based on feedback from the specific applied domain, (2) to fine tune and personalize those applications, and finally (3) to identify and tackle novel combinatorial and algorithmic problems, in some cases all the way to the development of novel software tools.
Cite as

Rudolf Ahlswede, Alberto Apostolico, and Vladimir I. Levenshtein. 06201 Executive Summary – Combinatorial and Algorithmic Foundations of Pattern and Association Discovery. In Combinatorial and Algorithmic Foundations of Pattern and Association Discovery. Dagstuhl Seminar Proceedings, Volume 6201, pp. 1-2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{ahlswede_et_al:DagSemProc.06201.2,
  author =	{Ahlswede, Rudolf and Apostolico, Alberto and Levenshtein, Vladimir I.},
  title =	{{06201 Executive Summary – Combinatorial and Algorithmic Foundations of Pattern and Association Discovery}},
  booktitle =	{Combinatorial and Algorithmic Foundations of Pattern and Association Discovery},
  pages =	{1--2},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6201},
  editor =	{Rudolf Ahlswede and Alberto Apostolico and Vladimir I. Levenshtein},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06201.2},
  URN =		{urn:nbn:de:0030-drops-7926},
  doi =		{10.4230/DagSemProc.06201.2},
  annote =	{Keywords: Data compression, pattern matching, pattern discovery, search, sorting, molecular biology, reconstruction, genome rearrangements}
}
Document
DOI: 10.4230/DagSemProc.06201.4
Non--binary error correcting codes with noiseless feedback, localized errors, or both

Authors: Rudolf Ahlswede, Christian Deppe, and Vladimir Lebedev

Published in: Dagstuhl Seminar Proceedings, Volume 6201, Combinatorial and Algorithmic Foundations of Pattern and Association Discovery (2006)

Abstract
We investigate non--binary error correcting codes with noiseless feedback, localized errors, or both. It turns out that the Hamming bound is a central concept. For block codes with feedback we present here a coding scheme based on an idea of erasions, which we call the {\bf rubber method}. It gives an optimal rate for big error correcting fraction $\tau$ ($>{1\over q}$) and infinitely many points on the Hamming bound for small $\tau$. We also consider variable length codes with all lengths bounded from above by $n$ and the end of a word carries the symbol $\Box$ and is thus recognizable by the decoder. For both, the $\Box$-model with feedback and the $\Box$-model with localized errors, the Hamming bound is the exact capacity curve for $\tau <1/2.$ Somewhat surprisingly, whereas with feedback the capacity curve coincides with the Hamming bound also for $1/2\leq \tau \leq 1$, in this range for localized errors the capacity curve equals 0. Also we give constructions for the models with both, feedback and localized errors.
Cite as

Rudolf Ahlswede, Christian Deppe, and Vladimir Lebedev. Non--binary error correcting codes with noiseless feedback, localized errors, or both. In Combinatorial and Algorithmic Foundations of Pattern and Association Discovery. Dagstuhl Seminar Proceedings, Volume 6201, pp. 1-4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{ahlswede_et_al:DagSemProc.06201.4,
  author =	{Ahlswede, Rudolf and Deppe, Christian and Lebedev, Vladimir},
  title =	{{Non--binary error correcting codes with noiseless feedback, localized errors, or both}},
  booktitle =	{Combinatorial and Algorithmic Foundations of Pattern and Association Discovery},
  pages =	{1--4},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6201},
  editor =	{Rudolf Ahlswede and Alberto Apostolico and Vladimir I. Levenshtein},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06201.4},
  URN =		{urn:nbn:de:0030-drops-7849},
  doi =		{10.4230/DagSemProc.06201.4},
  annote =	{Keywords: Error-correcting codes, localized errors, feedback, variable length codes}
}
Document
DOI: 10.4230/DagSemProc.06201.5
On the Monotonicity of the String Correction Factor for Words with Mismatches

Authors: Alberto Apostolico and Cinzia Pizzi

Published in: Dagstuhl Seminar Proceedings, Volume 6201, Combinatorial and Algorithmic Foundations of Pattern and Association Discovery (2006)

Abstract
The string correction factor is the term by which the probability of a word $w$ needs to be multiplied in order to account for character changes or ``errors'' occurring in at most $k$ arbitrary positions in that word. The behavior of this factor, as a function of $k$ and of the word length, has implications on the number of candidates that need to be considered and weighted when looking for subwords of a sequence that present unusually recurrent replicas within some bounded number of mismatches. Specifically, it is seen that over intervals of mono- or bi-tonicity for the correction factor, only some of the candidates need be considered. This mitigates the computation and leads to tables of over-represented words that are more compact to represent and inspect. In recent work, expectation and score monotonicity has been established for a number of cases of interest, under {em i.i.d.} probabilistic assumptions. The present paper reviews the cases of bi-tonic behavior for the correction factor, concentrating on the instance in which the question is still open.
Cite as

Alberto Apostolico and Cinzia Pizzi. On the Monotonicity of the String Correction Factor for Words with Mismatches. In Combinatorial and Algorithmic Foundations of Pattern and Association Discovery. Dagstuhl Seminar Proceedings, Volume 6201, pp. 1-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{apostolico_et_al:DagSemProc.06201.5,
  author =	{Apostolico, Alberto and Pizzi, Cinzia},
  title =	{{On the Monotonicity of the String Correction Factor for Words with Mismatches}},
  booktitle =	{Combinatorial and Algorithmic Foundations of Pattern and Association Discovery},
  pages =	{1--9},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6201},
  editor =	{Rudolf Ahlswede and Alberto Apostolico and Vladimir I. Levenshtein},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06201.5},
  URN =		{urn:nbn:de:0030-drops-7899},
  doi =		{10.4230/DagSemProc.06201.5},
  annote =	{Keywords: Pattern discovery, Motif, Over-represented word, Monotone score, Correction Factor}
}
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