5 Search Results for "Rudra, Atri"

Document
DOI: 10.4230/LIPIcs.ITCS.2025.14
The More the Merrier! On Total Coding and Lattice Problems and the Complexity of Finding Multicollisions

Authors: Huck Bennett, Surendra Ghentiyala, and Noah Stephens-Davidowitz

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

Abstract
We show a number of connections between two types of search problems: (1) the problem of finding an L-wise multicollision in the output of a function; and (2) the problem of finding two codewords in a code (or two vectors in a lattice) that are within distance d of each other. Specifically, we study these problems in the total regime, in which L and d are chosen so that such a solution is guaranteed to exist, though it might be hard to find. In more detail, we study the total search problem in which the input is a function 𝒞 : [A] → [B] (represented as a circuit) and the goal is to find L ≤ ⌈A/B⌉ distinct elements x_1,…, x_L ∈ A such that 𝒞(x_1) = ⋯ = 𝒞(x_L). The associated complexity classes Polynomial Multi-Pigeonhole Principle ((A,B)-PMPP^L) consist of all problems that reduce to this problem. We show close connections between (A,B)-PMPP^L and many celebrated upper bounds on the minimum distance of a code or lattice (and on the list-decoding radius). In particular, we show that the associated computational problems (i.e., the problem of finding two distinct codewords or lattice points that are close to each other) are in (A,B)-PMPP^L, with a more-or-less smooth tradeoff between the distance d and the parameters A, B, and L. These connections are particularly rich in the case of codes, in which case we show that multiple incomparable bounds on the minimum distance lie in seemingly incomparable complexity classes. Surprisingly, we also show that the computational problems associated with some bounds on the minimum distance of codes are actually hard for these classes (for codes represented by arbitrary circuits). In fact, we show that finding two vectors within a certain distance d is actually hard for the important (and well-studied) class PWPP = (B²,B)-PMPP² in essentially all parameter regimes for which an efficient algorithm is not known, so that our hardness results are essentially tight. In fact, for some d (depending on the block length, message length, and alphabet size), we obtain both hardness and containment. We therefore completely settle the complexity of this problem for such parameters and add coding problems to the short list of problems known to be complete for PWPP. We also study (A,B)-PMPP^L as an interesting family of complexity classes in its own right, and we uncover a rich structure. Specifically, we use recent techniques from the cryptographic literature on multicollision-resistant hash functions to (1) show inclusions of the form (A,B)-PMPP^L ⊆ (A',B')-PMPP^L' for certain non-trivial parameters; (2) black-box separations between such classes in different parameter regimes; and (3) a non-black-box proof that (A,B)-PMPP^L ∈ FP if (A',B')-PMPP^L' ∈ FP for yet another parameter regime. We also show that (A,B)-PMPP^L lies in the recently introduced complexity class Polynomial Long Choice for some parameters.
Cite as

Huck Bennett, Surendra Ghentiyala, and Noah Stephens-Davidowitz. The More the Merrier! On Total Coding and Lattice Problems and the Complexity of Finding Multicollisions. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 14:1-14:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bennett_et_al:LIPIcs.ITCS.2025.14,
  author =	{Bennett, Huck and Ghentiyala, Surendra and Stephens-Davidowitz, Noah},
  title =	{{The More the Merrier! On Total Coding and Lattice Problems and the Complexity of Finding Multicollisions}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{14:1--14:22},
  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.2025.14},
  URN =		{urn:nbn:de:0030-drops-226424},
  doi =		{10.4230/LIPIcs.ITCS.2025.14},
  annote =	{Keywords: Multicollisions, Error-correcting codes, Lattices}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.58
Sparse Recovery for Orthogonal Polynomial Transforms

Authors: Anna Gilbert, Albert Gu, Christopher Ré, Atri Rudra, and Mary Wootters

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
In this paper we consider the following sparse recovery problem. We have query access to a vector 𝐱 ∈ ℝ^N such that x̂ = 𝐅 𝐱 is k-sparse (or nearly k-sparse) for some orthogonal transform 𝐅. The goal is to output an approximation (in an 𝓁₂ sense) to x̂ in sublinear time. This problem has been well-studied in the special case that 𝐅 is the Discrete Fourier Transform (DFT), and a long line of work has resulted in sparse Fast Fourier Transforms that run in time O(k ⋅ polylog N). However, for transforms 𝐅 other than the DFT (or closely related transforms like the Discrete Cosine Transform), the question is much less settled. In this paper we give sublinear-time algorithms - running in time poly(k log(N)) - for solving the sparse recovery problem for orthogonal transforms 𝐅 that arise from orthogonal polynomials. More precisely, our algorithm works for any 𝐅 that is an orthogonal polynomial transform derived from Jacobi polynomials. The Jacobi polynomials are a large class of classical orthogonal polynomials (and include Chebyshev and Legendre polynomials as special cases), and show up extensively in applications like numerical analysis and signal processing. One caveat of our work is that we require an assumption on the sparsity structure of the sparse vector, although we note that vectors with random support have this property with high probability. Our approach is to give a very general reduction from the k-sparse sparse recovery problem to the 1-sparse sparse recovery problem that holds for any flat orthogonal polynomial transform; then we solve this one-sparse recovery problem for transforms derived from Jacobi polynomials. Frequently, sparse FFT algorithms are described as implementing such a reduction; however, the technical details of such works are quite specific to the Fourier transform and moreover the actual implementations of these algorithms do not use the 1-sparse algorithm as a black box. In this work we give a reduction that works for a broad class of orthogonal polynomial families, and which uses any 1-sparse recovery algorithm as a black box.
Cite as

Anna Gilbert, Albert Gu, Christopher Ré, Atri Rudra, and Mary Wootters. Sparse Recovery for Orthogonal Polynomial Transforms. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 58:1-58:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gilbert_et_al:LIPIcs.ICALP.2020.58,
  author =	{Gilbert, Anna and Gu, Albert and R\'{e}, Christopher and Rudra, Atri and Wootters, Mary},
  title =	{{Sparse Recovery for Orthogonal Polynomial Transforms}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{58:1--58:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.58},
  URN =		{urn:nbn:de:0030-drops-124653},
  doi =		{10.4230/LIPIcs.ICALP.2020.58},
  annote =	{Keywords: Orthogonal polynomials, Jacobi polynomials, sublinear algorithms, sparse recovery}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.7
Faster Algorithms for All-Pairs Bounded Min-Cuts

Authors: Amir Abboud, Loukas Georgiadis, Giuseppe F. Italiano, Robert Krauthgamer, Nikos Parotsidis, Ohad Trabelsi, Przemysław Uznański, and Daniel Wolleb-Graf

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

Abstract
The All-Pairs Min-Cut problem (aka All-Pairs Max-Flow) asks to compute a minimum s-t cut (or just its value) for all pairs of vertices s,t. We study this problem in directed graphs with unit edge/vertex capacities (corresponding to edge/vertex connectivity). Our focus is on the k-bounded case, where the algorithm has to find all pairs with min-cut value less than k, and report only those. The most basic case k=1 is the Transitive Closure (TC) problem, which can be solved in graphs with n vertices and m edges in time O(mn) combinatorially, and in time O(n^{omega}) where omega<2.38 is the matrix-multiplication exponent. These time bounds are conjectured to be optimal. We present new algorithms and conditional lower bounds that advance the frontier for larger k, as follows: - A randomized algorithm for vertex capacities that runs in time {O}((nk)^{omega}). This is only a factor k^omega away from the TC bound, and nearly matches it for all k=n^{o(1)}. - Two deterministic algorithms for edge capacities (which is more general) that work in DAGs and further reports a minimum cut for each pair. The first algorithm is combinatorial (does not involve matrix multiplication) and runs in time {O}(2^{{O}(k^2)}* mn). The second algorithm can be faster on dense DAGs and runs in time {O}((k log n)^{4^{k+o(k)}}* n^{omega}). Previously, Georgiadis et al. [ICALP 2017], could match the TC bound (up to n^{o(1)} factors) only when k=2, and now our two algorithms match it for all k=o(sqrt{log n}) and k=o(log log n). - The first super-cubic lower bound of n^{omega-1-o(1)} k^2 time under the 4-Clique conjecture, which holds even in the simplest case of DAGs with unit vertex capacities. It improves on the previous (SETH-based) lower bounds even in the unbounded setting k=n. For combinatorial algorithms, our reduction implies an n^{2-o(1)} k^2 conditional lower bound. Thus, we identify new settings where the complexity of the problem is (conditionally) higher than that of TC. Our three sets of results are obtained via different techniques. The first one adapts the network coding method of Cheung, Lau, and Leung [SICOMP 2013] to vertex-capacitated digraphs. The second set exploits new insights on the structure of latest cuts together with suitable algebraic tools. The lower bounds arise from a novel reduction of a different structure than the SETH-based constructions.
Cite as

Amir Abboud, Loukas Georgiadis, Giuseppe F. Italiano, Robert Krauthgamer, Nikos Parotsidis, Ohad Trabelsi, Przemysław Uznański, and Daniel Wolleb-Graf. Faster Algorithms for All-Pairs Bounded Min-Cuts. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{abboud_et_al:LIPIcs.ICALP.2019.7,
  author =	{Abboud, Amir and Georgiadis, Loukas and Italiano, Giuseppe F. and Krauthgamer, Robert and Parotsidis, Nikos and Trabelsi, Ohad and Uzna\'{n}ski, Przemys{\l}aw and Wolleb-Graf, Daniel},
  title =	{{Faster Algorithms for All-Pairs Bounded Min-Cuts}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{7:1--7:15},
  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.2019.7},
  URN =		{urn:nbn:de:0030-drops-105833},
  doi =		{10.4230/LIPIcs.ICALP.2019.7},
  annote =	{Keywords: All-pairs min-cut, k-reachability, network coding, Directed graphs, fine-grained complexity}
}
Document
DOI: 10.4230/LIPIcs.STACS.2012.230
Efficiently Decodable Compressed Sensing by List-Recoverable Codes and Recursion

Authors: Hung Q. Ngo, Ely Porat, and Atri Rudra

Published in: LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

Abstract
We present two recursive techniques to construct compressed sensing schemes that can be "decoded" in sub-linear time. The first technique is based on the well studied code composition method called code concatenation where the "outer" code has strong list recoverability properties. This technique uses only one level of recursion and critically uses the power of list recovery. The second recursive technique is conceptually similar, and has multiple recursion levels. The following compressed sensing results are obtained using these techniques: - Strongly explicit efficiently decodable l_1/l_1 compressed sensing matrices: We present a strongly explicit ("for all") compressed sensing measurement matrix with O(d^2log^2 n) measurements that can output near-optimal d-sparse approximations in time poly(d log n). - Near-optimal efficiently decodable l_1/l_1 compressed sensing matrices for non-negative signals: We present two randomized constructions of ("for all") compressed sensing matrices with near optimal number of measurements: O(d log n loglog_d n) and O_{m,s}(d^{1+1/s} log n (log^(m) n)^s), respectively, for any integer parameters s,m>=1. Both of these constructions can output near optimal d-sparse approximations for non-negative signals in time poly(d log n). To the best of our knowledge, none of the results are dominated by existing results in the literature.
Cite as

Hung Q. Ngo, Ely Porat, and Atri Rudra. Efficiently Decodable Compressed Sensing by List-Recoverable Codes and Recursion. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 230-241, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{ngo_et_al:LIPIcs.STACS.2012.230,
  author =	{Ngo, Hung Q. and Porat, Ely and Rudra, Atri},
  title =	{{Efficiently Decodable Compressed Sensing by List-Recoverable Codes and Recursion}},
  booktitle =	{29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
  pages =	{230--241},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-35-4},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{14},
  editor =	{D\"{u}rr, Christoph and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.230},
  URN =		{urn:nbn:de:0030-drops-34011},
  doi =		{10.4230/LIPIcs.STACS.2012.230},
  annote =	{Keywords: Compressed Sensing, Sub-Linear Time Decoding, List-Recoverable Codes}
}
Document
DOI: 10.4230/LIPIcs.STACS.2011.428
Polynomial Fitting of Data Streams with Applications to Codeword Testing

Authors: Andrew McGregor, Atri Rudra, and Steve Uurtamo

Published in: LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

Abstract
Given a stream of $(x,y)$ points, we consider the problem of finding univariate polynomials that best fit the data. Over finite fields, this problem encompasses the well-studied problem of decoding Reed-Solomon codes while over the reals it corresponds to the well-studied polynomial regression problem. We present one-pass algorithms for two natural problems: i) find the polynomial of a given degree $k$ that minimizes the error and ii) find the polynomial of smallest degree that interpolates through the points with at most a given error bound. We consider a range of error models including the average error per point, the maximum error, and the number of points that are not fitted exactly. Many of our results apply to both the reals and finite fields. As a consequence we also solve an open question regarding the tolerant testing of codes in the data stream model.
Cite as

Andrew McGregor, Atri Rudra, and Steve Uurtamo. Polynomial Fitting of Data Streams with Applications to Codeword Testing. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 428-439, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{mcgregor_et_al:LIPIcs.STACS.2011.428,
  author =	{McGregor, Andrew and Rudra, Atri and Uurtamo, Steve},
  title =	{{Polynomial Fitting of Data Streams with Applications to Codeword Testing}},
  booktitle =	{28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)},
  pages =	{428--439},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-25-5},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{9},
  editor =	{Schwentick, Thomas and D\"{u}rr, Christoph},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.428},
  URN =		{urn:nbn:de:0030-drops-30322},
  doi =		{10.4230/LIPIcs.STACS.2011.428},
  annote =	{Keywords: Streaming, Polynomial Interpolation, Polynomial Regression}
}
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