arXiv subjects

Ryan Gabrys

Publications and source records attributed to Ryan Gabrys.

At least 37 records · Page 2Linked to original sources

arXiv2023

Storage codes and recoverable systems on lines and grids

A storage code is an assignment of symbols to the vertices of a connected graph $G(V,E)$ with the property that the value of each vertex is a function of the values of its neighbors, or more generally, of a certain neighborhood of the vertex in $G$. In this work we introduce a new construction method of storage codes, enabling one to construct new codes from known ones via an interleaving procedure driven by resolvable designs. We also study storage codes on $\mathbb Z$ and ${\mathbb Z}^2$ (lines and grids), finding closed-form expressions for the capacity of several one and two-dimensional systems depending on their recovery set, using connections between storage codes, graphs, anticodes, and difference-avoiding sets.

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arXiv2023

Accelerating Polarization via Alphabet Extension

Polarization is an unprecedented coding technique in that it not only achieves channel capacity, but also does so at a faster speed of convergence than any other coding technique. This speed is measured by the ``scaling exponent'' and its importance is three-fold. Firstly, estimating the scaling exponent is challenging and demands a deeper understanding of the dynamics of communication channels. Secondly, scaling exponents serve as a benchmark for different variants of polar codes that helps us select the proper variant for real-life applications. Thirdly, the need to optimize for the scaling exponent sheds light on how to reinforce the design of polar codes. In this paper, we generalize the binary erasure channel (BEC), the simplest communication channel and the protagonist of many coding theory studies, to the ``tetrahedral erasure channel'' (TEC). We then invoke Mori--Tanaka's $2 \times 2$ matrix over GF$(4)$ to construct polar codes over TEC. Our main contribution is showing that the dynamic of TECs converges to an almost--one-parameter family of channels, which then leads to an upper bound of $3.328$ on the scaling exponent. This is the first non-binary matrix whose scaling exponent is upper-bounded. It also polarizes BEC faster than all known binary matrices up to $23 \times 23$ in size. Our result indicates that expanding the alphabet is a more effective and practical alternative to enlarging the matrix in order to achieve faster polarization.

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arXiv2023

Finding a Burst of Positives via Nonadaptive Semiquantitative Group Testing

Motivated by testing for pathogenic diseases we consider a new nonadaptive group testing problem for which: (1) positives occur within a burst, capturing the fact that infected test subjects often come in clusters, and (2) that the test outcomes arise from semiquantitative measurements that provide coarse information about the number of positives in any tested group. Our model generalizes prior work on detecting a single burst of positives with classical group testing[1] as well as work on semiquantitative group testing (SQGT)[2]. Specifically, we study the setting where the burst-length $\ell$ is known and the semiquantitative tests provide potentially nonuniform estimates on the number of positives in a test group. The estimates represent the index of a quantization bin containing the (exact) total number of positives, for arbitrary thresholds $η_1,\dots,η_s$. Interestingly, we show that the minimum number of tests needed for burst identification is essentially only a function of the largest threshold $η_s$. In this context, our main result is an order-optimal test scheme that can recover any burst of length $\ell$ using roughly $\frac{\ell}{2η_s}+\log_{s+1}(n)$ measurements. This suggests that a large saturation level $η_s$ is more important than finely quantized information when dealing with bursts. We also provide results for related modeling assumptions and specialized choices of thresholds.

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arXiv2023

Transition Waste Optimization for Coded Elastic Computing

Distributed computing, in which a resource-intensive task is divided into subtasks and distributed among different machines, plays a key role in solving large-scale problems. Coded computing is a recently emerging paradigm where redundancy for distributed computing is introduced to alleviate the impact of slow machines (stragglers) on the completion time. We investigate coded computing solutions over elastic resources, where the set of available machines may change in the middle of the computation. This is motivated by recently available services in the cloud computing industry (e.g., EC2 Spot, Azure Batch) where low-priority virtual machines are offered at a fraction of the price of the on-demand instances but can be preempted on short notice. Our contributions are three-fold. We first introduce a new concept called transition waste that quantifies the number of tasks existing machines must abandon or take over when a machine joins/leaves. We then develop an efficient method to minimize the transition waste for the cyclic task allocation scheme recently proposed in the literature (Yang et al. ISIT'19). Finally, we establish a novel solution based on finite geometry achieving zero transition wastes given that the number of active machines varies within a fixed range.

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arXiv2022

Non-binary Codes for Correcting a Burst of at Most t Deletions

The problem of correcting deletions has received significant attention, partly because of the prevalence of these errors in DNA data storage. In this paper, we study the problem of correcting a consecutive burst of at most $t$ deletions in non-binary sequences. We first propose a non-binary code correcting a burst of at most 2 deletions for $q$-ary alphabets. Afterwards, we extend this result to the case where the length of the burst can be at most $t$ where $t$ is a constant. Finally, we consider the setup where the sequences that are transmitted are permutations. The proposed codes are the largest known for their respective parameter regimes.

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arXiv2022

Low-redundancy codes for correcting multiple short-duplication and edit errors

Due to its higher data density, longevity, energy efficiency, and ease of generating copies, DNA is considered a promising storage technology for satisfying future needs. However, a diverse set of errors including deletions, insertions, duplications, and substitutions may arise in DNA at different stages of data storage and retrieval. The current paper constructs error-correcting codes for simultaneously correcting short (tandem) duplications and at most $p$ edits, where a short duplication generates a copy of a substring with length $\leq 3$ and inserts the copy following the original substring, and an edit is a substitution, deletion, or insertion. Compared to the state-of-the-art codes for duplications only, the proposed codes correct up to $p$ edits (in addition to duplications) at the additional cost of roughly $8p(\log_q n)(1+o(1))$ symbols of redundancy, thus achieving the same asymptotic rate, where $q\ge 4$ is the alphabet size and $p$ is a constant. Furthermore, the time complexities of both the encoding and decoding processes are polynomial when $p$ is a constant with respect to the code length.

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arXiv2022

The Gapped $k$-Deck Problem

The $k$-deck problem is concerned with finding the smallest positive integer $S(k)$ such that there exist at least two strings of length $S(k)$ that share the same $k$-deck, i.e., the multiset of subsequences of length $k$. We introduce the new problem of gapped $k$-deck reconstruction: For a given gap parameter $s$, we seek the smallest positive integer $G_s(k)$ such that there exist at least two distinct strings of length $G_s(k)$ that cannot be distinguished based on a "gapped" set of $k$-subsequences. The gap constraint requires the elements in the subsequences to be at least $s$ positions apart within the original string. Our results are as follows. First, we show how to construct sequences sharing the same $2$-gapped $k$-deck using a nontrivial modification of the recursive Morse-Thue string construction procedure. This establishes the first known constructive upper bound on $G_2(k)$. Second, we further improve this bound using the approach by Dudik and Schulman.

math.CO↗

arXiv2022

Balanced and Swap-Robust Trades for Dynamical Distributed Storage

Trades, introduced by Hedayat, are two sets of blocks of elements which may be exchanged (traded) without altering the counts of certain subcollections of elements within their constituent blocks. They are of importance in applications where certain combinations of elements dynamically become prohibited from being placed in the same group of elements, since in this case one can trade the offending blocks with allowed ones. This is particularly the case in distributed storage systems, where due to privacy and other constraints, data of some groups of users cannot be stored together on the same server. We introduce a new class of balanced trades, important for access balancing of servers, and perturbation resilient balanced trades, important for studying the stability of server access frequencies with respect to changes in data popularity. The constructions and bounds on our new trade schemes rely on specialized selections of defining sets in minimal trades and number-theoretic analyses.

math.CO↗

arXiv2022

Sub-4.7 Scaling Exponent of Polar Codes

Polar code visibly approaches channel capacity in practice and is thereby a constituent code of the 5G standard. Compared to low-density parity-check code, however, the performance of short-length polar code has rooms for improvement that could hinder its adoption by a wider class of applications. As part of the program that addresses the performance issue at short length, it is crucial to understand how fast binary memoryless symmetric channels polarize. A number, called scaling exponent, was defined to measure the speed of polarization and several estimates of the scaling exponent were given in literature. As of 2022, the tightest overestimate is 4.714 made by Mondelli, Hassani, and Urbanke in 2015. We lower the overestimate to 4.63.

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arXiv2022

Tropical Group Testing

Polymerase chain reaction (PCR) testing is the gold standard for diagnosing COVID-19. PCR amplifies the virus DNA 40 times to produce measurements of viral loads that span seven orders of magnitude. Unfortunately, the outputs of these tests are imprecise and therefore quantitative group testing methods, which rely on precise measurements, are not applicable. Motivated by the ever-increasing demand to identify individuals infected with SARS-CoV-19, we propose a new model that leverages tropical arithmetic to characterize the PCR testing process. Our proposed framework, termed tropical group testing, overcomes existing limitations of quantitative group testing by allowing for imprecise test measurements. In many cases, some of which are highlighted in this work, tropical group testing is provably more powerful than traditional binary group testing in that it require fewer tests than classical approaches, while additionally providing a mechanism to identify the viral load of each infected individual. It is also empirically stronger than related works that have attempted to combine PCR, quantitative group testing, and compressed sensing.

q-bio.QM↗

arXiv2021

Beyond Single-Deletion Correcting Codes: Substitutions and Transpositions

We consider the problem of designing low-redundancy codes in settings where one must correct deletions in conjunction with substitutions or adjacent transpositions; a combination of errors that is usually observed in DNA-based data storage. One of the most basic versions of this problem was settled more than 50 years ago by Levenshtein, or one substitution, with nearly optimal redundancy. However, this approach fails to extend to many simple and natural variations of the binary single-edit error setting. In this work, we make progress on the code design problem above in three such variations: We construct linear-time encodable and decodable length-$n$ non-binary codes correcting a single edit error with nearly optimal redundancy $\log n+O(\log\log n)$, providing an alternative simpler proof of a result by Cai, Chee, Gabrys, Kiah, and Nguyen (IEEE Trans. Inf. Theory 2021). This is achieved by employing what we call weighted VT sketches, a notion that may be of independent interest. We construct linear-time encodable and list-decodable binary codes with list-size $2$ for one deletion and one substitution with redundancy $4\log n+O(\log\log n)$. This matches the existential bound up to an $O(\log\log n)$ additive term. We show the existence of a binary code correcting one deletion or one adjacent transposition with nearly optimal redundancy $\log n+O(\log\log n)$.

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arXiv2021

Reconstruction of Sets of Strings from Prefix/Suffix Compositions

The problem of reconstructing strings from substring information has found many applications due to its importance in genomic data sequencing and DNA- and polymer-based data storage. One practically important and challenging paradigm requires reconstructing mixtures of strings based on the union of compositions of their prefixes and suffixes, generated by mass spectrometry devices. We describe new coding methods that allow for unique joint reconstruction of subsets of strings selected from a code and provide upper and lower bounds on the asymptotic rate of the underlying codebooks. Our code constructions combine properties of binary Bh and Dyck strings and that can be extended to accommodate missing substrings in the pool. As auxiliary results, we obtain the first known bounds on binary Bh sequences for arbitrary even parameters h, and also describe various error models inherent to mass spectrometry analysis. This paper contains a correction of the prior work by the authors, published in [24]. In particular, the bounds on the prefix codes are now corrected.

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arXiv2021

Coding for Polymer-Based Data Storage

Motivated by polymer-based data-storage platforms that use chains of binary synthetic polymers as the recording media and read the content via tandem mass spectrometers, we propose a new family of codes that allows for both unique string reconstruction and correction of multiple mass errors. We consider two approaches: The first approach pertains to asymmetric errors and it is based on introducing redundancy that scales linearly with the number of errors and logarithmically with the length of the string. The construction allows for the string to be uniquely reconstructed based only on its erroneous substring composition multiset. The key idea behind our unique reconstruction approach is to interleave (shifted) Catalan-Bertrand paths with arbitrary binary strings and "reflect" them so as to force prefixes and suffixes of the same length to have different weights. The asymptotic code rate of the scheme is one, and decoding is accomplished via a simplified version of the backtracking algorithm used for the Turnpike problem. For symmetric errors, we use a polynomial characterization of the mass information and adapt polynomial evaluation code constructions for this setting. In the process, we develop new efficient decoding algorithms for a constant number of composition errors and show that the redundancy of the scheme scales quadratically with the number of errors and logarithmically with the codelength.

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arXiv2021

Repeat-Free Codes

In this paper we consider the problem of encoding data into \textit{repeat-free} sequences in which sequences are imposed to contain any $k$-tuple at most once (for predefined $k$). First, the capacity of the repeat-free constraint are calculated. Then, an efficient algorithm, which uses two bits of redundancy, is presented to encode length-$n$ sequences for $k=2+2\log (n)$. This algorithm is then improved to support any value of $k$ of the form $k=a\log (n)$, for $1<a$, while its redundancy is $o(n)$. We also calculate the capacity of repeat-free sequences when combined with local constraints which are given by a constrained system, and the capacity of multi-dimensional repeat-free codes.

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arXiv2021

AC-DC: Amplification Curve Diagnostics for Covid-19 Group Testing

The first part of the paper presents a review of the gold-standard testing protocol for Covid-19, real-time, reverse transcriptase PCR, and its properties and associated measurement data such as amplification curves that can guide the development of appropriate and accurate adaptive group testing protocols. The second part of the paper is concerned with examining various off-the-shelf group testing methods for Covid-19 and identifying their strengths and weaknesses for the application at hand. The third part of the paper contains a collection of new analytical results for adaptive semiquantitative group testing with probabilistic and combinatorial priors, including performance bounds, algorithmic solutions, and noisy testing protocols. The probabilistic setting is of special importance as it is designed to be simple to implement by nonexperts and handle heavy hitters. The worst-case paradigm extends and improves upon prior work on semiquantitative group testing with and without specialized PCR noise models.

q-bio.QM↗

arXiv2021

Semiquantitative Group Testing in at Most Two Rounds

Semiquantitative group testing (SQGT) is a pooling method in which the test outcomes represent bounded intervals for the number of defectives. Alternatively, it may be viewed as an adder channel with quantized outputs. SQGT represents a natural choice for Covid-19 group testing as it allows for a straightforward interpretation of the cycle threshold values produced by polymerase chain reactions (PCR). Prior work on SQGT did not address the need for adaptive testing with a small number of rounds as required in practice. We propose conceptually simple methods for 2-round and nonadaptive SQGT that significantly improve upon existing schemes by using ideas on nonbinary measurement matrices based on expander graphs and list-disjunct matrices.

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arXiv2020

Reconstructing Mixtures of Coded Strings from Prefix and Suffix Compositions

The problem of string reconstruction from substring information has found many applications due to its relevance in DNA- and polymer-based data storage. One practically important and challenging paradigm requires reconstructing mixtures of strings based on the union of compositions of their prefixes and suffixes, generated by mass spectrometry readouts. We describe new coding methods that allow for unique joint reconstruction of subsets of strings selected from a code and provide matching upper and lower bounds on the asymptotic rate of the underlying codebooks. Under certain mild constraints on the problem parameters, one can show that the largest possible rate of a codebook that allows for all subcollections of $\leq h$ codestrings to be uniquely reconstructable from the prefix-suffix information equals $1/h$.

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arXiv2020

Mass Error-Correction Codes for Polymer-Based Data Storage

We consider the problem of correcting mass readout errors in information encoded in binary polymer strings. Our work builds on results for string reconstruction problems using composition multisets [Acharya et al., 2015] and the unique string reconstruction framework proposed in [Pattabiraman et al., 2019]. Binary polymer-based data storage systems [Laure et al., 2016] operate by designing two molecules of significantly different masses to represent the symbols $\{0,1\}$ and perform readouts through noisy tandem mass spectrometry. Tandem mass spectrometers fragment the strings to be read into shorter substrings and only report their masses, often with errors due to imprecise ionization. Modeling the fragmentation process output in terms of composition multisets allows for designing asymptotically optimal codes capable of unique reconstruction and the correction of a single mass error [Pattabiraman et al., 2019] through the use of derivatives of Catalan paths. Nevertheless, no solutions for multiple-mass error-corrections are currently known. Our work addresses this issue by describing the first multiple-error correction codes that use the polynomial factorization approach for the Turnpike problem [Skiena et al., 1990] and the related factorization described in [Acharya et al., 2015]. Adding Reed-Solomon type coding redundancy into the corresponding polynomials allows for correcting $t$ mass errors in polynomial time using $t^2\, \log\,k$ redundant bits, where $k$ is the information string length. The redundancy can be improved to $\log\,k + t$. However, no decoding algorithm that runs polynomial-time in both $t$ and $n$ for this scheme are currently known, where $n$ is the length of the coded string.

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