4 Search Results for "Sha, Harry"

Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.34
Eigenvalue Bounds for Symmetric Markov Chains on Multislices with Applications

Authors: Prashanth Amireddy, Amik Raj Behera, Srikanth Srinivasan, and Madhu Sudan

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
We consider random walks on "balanced multislices" of any "grid" that respects the "symmetries" of the grid, and show that a broad class of such walks are good spectral expanders. (A grid is a set of points of the form 𝒮ⁿ for finite 𝒮, and a balanced multi-slice is the subset that contains an equal number of coordinates taking every value in 𝒮. A walk respects symmetries if the probability of going from u = (u_1,…,u_n) to v = (v_1,…,v_n) is invariant under simultaneous permutations of the coordinates of u and v.) Our main theorem shows that, under some technical conditions, every such walk where a single step leads to an almost 𝒪(1)-wise independent distribution on the next state, conditioned on the previous state, satisfies a non-trivially small singular value bound. We give two applications of our theorem to error-correcting codes: (1) We give an analog of the Ore-DeMillo-Lipton-Schwartz-Zippel lemma for polynomials, and junta-sums, over balanced multislices. (2) We also give a local list-correction algorithm for d-junta-sums mapping an arbitrary grid 𝒮ⁿ to an Abelian group, correcting from a near-optimal (1/|𝒮|^d - ε) fraction of errors for every ε > 0, where a d-junta-sum is a sum of (arbitrarily many) d-juntas (and a d-junta is a function that depends on only d of the n variables). Our proofs are obtained by exploring the representation theory of the symmetric group and merging it with some careful spectral analysis.
Cite as

Prashanth Amireddy, Amik Raj Behera, Srikanth Srinivasan, and Madhu Sudan. Eigenvalue Bounds for Symmetric Markov Chains on Multislices with Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 34:1-34:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{amireddy_et_al:LIPIcs.APPROX/RANDOM.2025.34,
  author =	{Amireddy, Prashanth and Behera, Amik Raj and Srinivasan, Srikanth and Sudan, Madhu},
  title =	{{Eigenvalue Bounds for Symmetric Markov Chains on Multislices with Applications}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{34:1--34:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.34},
  URN =		{urn:nbn:de:0030-drops-244004},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.34},
  annote =	{Keywords: Markov Chains, Random Walk, Multislices, Representation Theory of Symmetric Group, Local Correction, Low-degree Polynomials, Polynomial Distance Lemma}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.67
Error-Correcting Graph Codes

Authors: Swastik Kopparty, Aditya Potukuchi, and Harry Sha

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

Abstract
In this paper, we construct Error-Correcting Graph Codes. An error-correcting graph code of distance δ is a family C of graphs, on a common vertex set of size n, such that if we start with any graph in C, we would have to modify the neighborhoods of at least δ n vertices in order to obtain some other graph in C. This is a natural graph generalization of the standard Hamming distance error-correcting codes for binary strings. Yohananov and Yaakobi were the first to construct codes in this metric. We extend their work by showing 1) Combinatorial results determining the optimal rate vs distance trade-off nonconstructively. 2) Graph code analogues of Reed-Solomon codes and code concatenation, leading to positive distance codes for all rates and positive rate codes for all distances. 3) Graph code analogues of dual-BCH codes, yielding large codes with distance δ = 1-o(1). This gives an explicit "graph code of Ramsey graphs". Several recent works, starting with the paper of Alon, Gujgiczer, Körner, Milojević, and Simonyi, have studied more general graph codes; where the symmetric difference between any two graphs in the code is required to have some desired property. Error-correcting graph codes are a particularly interesting instantiation of this concept.
Cite as

Swastik Kopparty, Aditya Potukuchi, and Harry Sha. Error-Correcting Graph Codes. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 67:1-67:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kopparty_et_al:LIPIcs.ITCS.2025.67,
  author =	{Kopparty, Swastik and Potukuchi, Aditya and Sha, Harry},
  title =	{{Error-Correcting Graph Codes}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{67:1--67:20},
  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.2025.67},
  URN =		{urn:nbn:de:0030-drops-226950},
  doi =		{10.4230/LIPIcs.ITCS.2025.67},
  annote =	{Keywords: Graph codes, explicit construction, concatenation codes, tensor codes}
}
Document
Vision
DOI: 10.4230/TGDK.1.1.6
Towards Ordinal Data Science

Authors: Gerd Stumme, Dominik Dürrschnabel, and Tom Hanika

Published in: TGDK, Volume 1, Issue 1 (2023): Special Issue on Trends in Graph Data and Knowledge. Transactions on Graph Data and Knowledge, Volume 1, Issue 1

Abstract
Order is one of the main instruments to measure the relationship between objects in (empirical) data. However, compared to methods that use numerical properties of objects, the amount of ordinal methods developed is rather small. One reason for this is the limited availability of computational resources in the last century that would have been required for ordinal computations. Another reason - particularly important for this line of research - is that order-based methods are often seen as too mathematically rigorous for applying them to real-world data. In this paper, we will therefore discuss different means for measuring and ‘calculating’ with ordinal structures - a specific class of directed graphs - and show how to infer knowledge from them. Our aim is to establish Ordinal Data Science as a fundamentally new research agenda. Besides cross-fertilization with other cornerstone machine learning and knowledge representation methods, a broad range of disciplines will benefit from this endeavor, including, psychology, sociology, economics, web science, knowledge engineering, scientometrics.
Cite as

Gerd Stumme, Dominik Dürrschnabel, and Tom Hanika. Towards Ordinal Data Science. In Special Issue on Trends in Graph Data and Knowledge. Transactions on Graph Data and Knowledge (TGDK), Volume 1, Issue 1, pp. 6:1-6:39, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Article{stumme_et_al:TGDK.1.1.6,
  author =	{Stumme, Gerd and D\"{u}rrschnabel, Dominik and Hanika, Tom},
  title =	{{Towards Ordinal Data Science}},
  journal =	{Transactions on Graph Data and Knowledge},
  pages =	{6:1--6:39},
  ISSN =	{2942-7517},
  year =	{2023},
  volume =	{1},
  number =	{1},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/TGDK.1.1.6},
  URN =		{urn:nbn:de:0030-drops-194801},
  doi =		{10.4230/TGDK.1.1.6},
  annote =	{Keywords: Order relation, data science, relational theory of measurement, metric learning, general algebra, lattices, factorization, approximations and heuristics, factor analysis, visualization, browsing, explainability}
}
Document
DOI: 10.4230/LIPIcs.SAT.2022.9
A Generalization of the Satisfiability Coding Lemma and Its Applications

Authors: Milan Mossé, Harry Sha, and Li-Yang Tan

Published in: LIPIcs, Volume 236, 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)

Abstract
The seminal Satisfiability Coding Lemma of Paturi, Pudlák, and Zane is a coding scheme for satisfying assignments of k-CNF formulas. We generalize it to give a coding scheme for implicants and use this generalized scheme to establish new structural and algorithmic properties of prime implicants of k-CNF formulas. Our first application is a near-optimal bound of n⋅ 3^{n(1-Ω(1/k))} on the number of prime implicants of any n-variable k-CNF formula. This resolves an open problem from the Ph.D. thesis of Talebanfard, who proved such a bound for the special case of constant-read k-CNF formulas. Our proof is algorithmic in nature, yielding an algorithm for computing the set of all prime implicants - the Blake Canonical Form - of a given k-CNF formula. The problem of computing the Blake Canonical Form of a given function is a classic one, dating back to Quine, and our work gives the first non-trivial algorithm for k-CNF formulas.
Cite as

Milan Mossé, Harry Sha, and Li-Yang Tan. A Generalization of the Satisfiability Coding Lemma and Its Applications. In 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 236, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{mosse_et_al:LIPIcs.SAT.2022.9,
  author =	{Moss\'{e}, Milan and Sha, Harry and Tan, Li-Yang},
  title =	{{A Generalization of the Satisfiability Coding Lemma and Its Applications}},
  booktitle =	{25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)},
  pages =	{9:1--9:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-242-6},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{236},
  editor =	{Meel, Kuldeep S. and Strichman, Ofer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2022.9},
  URN =		{urn:nbn:de:0030-drops-166837},
  doi =		{10.4230/LIPIcs.SAT.2022.9},
  annote =	{Keywords: Prime Implicants, Satisfiability Coding Lemma, Blake Canonical Form, k-SAT}
}
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