Reconstruction of Strings from their Substrings Spectrum

Marcovich, Sagi; Yaakobi, Eitan

doi:10.1109/isit44484.2020.9174113

Cited by 10 publications

(7 citation statements)

References 21 publications

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“…These results are specifically discussed in remark 6, remark 7, and remark 9. Our work extends the growing list of recent work in string reconstruction problems [14]- [20].…”

Section: Introductionmentioning

confidence: 54%

A New Algebraic Approach for String Reconstruction from Substring Compositions

Gupta¹,

Mahdavifar²

2022

Preprint

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We consider the problem of binary string reconstruction from the multiset of its substring compositions, i.e., referred to as the substring composition multiset, first introduced and studied by Acharya et al. We introduce a new algorithm for the problem of string reconstruction from its substring composition multiset which relies on the algebraic properties of the equivalent bivariate polynomial formulation of the problem. We then characterize specific algebraic conditions for the binary string to be reconstructed that guarantee the algorithm does not require any backtracking through the reconstruction, and, consequently, the time complexity is bounded polynomially. More specifically, in the case of no backtracking, our algorithm has a time complexity of Opn 2 q compared to the algorithm by Acharya et al., which has a time complexity of Opn 2 log nq, where n is the length of the binary string. Furthermore, it is shown that larger sets of binary strings are uniquely reconstructable by the new algorithm and without the need for backtracking leading to codebooks of reconstruction codes that are larger, by a linear factor in size, compared to the previously known construction by Pattabiraman et al., while having Opn 2 q reconstruction complexity.

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“…These results are specifically discussed in remark 6, remark 7, and remark 9. Our work extends the growing list of recent work in string reconstruction problems [14]- [20].…”

Section: Introductionmentioning

confidence: 54%

A New Algebraic Approach for String Reconstruction from Substring Compositions

Gupta¹,

Mahdavifar²

2022

Preprint

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“…Table 1 reports the data for human chromosome 1 up to k = 40. It appears remarkable that already in the range [28][29][30][31][32][33][34][35][36][37][38][39][40] for the k values, the genomic coverage of the k-segmentation is almost total (over positions where there is no N), and almost all the k-mers present in the chromosome are involved by kspectral maximal segments (see the normalized cardinality of set U k ), that is, they have the property to be univocally elongated in the k-spectrum. Essentially, we see that for k = 27 human chromosome 1 may be covered by 4 millions of relatively short (80 bp long) spectral maximal segments, which have of course a high value of multiplicity in the segmentation, whereas for k = 44 about half of the spectral maximal segments (1.985.225) of average length 157 cover the chromosome.…”

Section: Computational Resultsmentioning

confidence: 99%

“…In sequence analysis, the term "spectrum" is used in many contexts and with several meanings (see for example, [30,32,21,19,31]), in particular to tackle the problem of reconstructing a string from the compositions of its substrings. Other terms are used with a similar meaning, as bags of words, where a k-spectrum is represented by a multiplicity vector, where the i-component gives the multiplicity of the k-mer of position i in some prefixed (for example lexicographic) order [23].…”

Section: Introductionmentioning

confidence: 99%

“…Reconstruction of a string from a few substrings was considered in [14,30] as well as the case where, for each substring, only the composition is given, neither the order of the symbols within it nor the substring's location in the original string. For example, in [31] reconstruction of strings based upon their error-prone substrings spectrum is developed, while the noisy setup of this problem was first studied by Gabrys and Milenkovic [19].…”

Section: Introductionmentioning

confidence: 99%

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Spectral concepts in genome informational analysis

Bonnici¹,

Franco²,

Manca³

2021

Preprint

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The concept of k-spectrum for genomes is here investigated as a basic tool to analyze genomes. Related spectral notions based on k-mers are introduced with some related mathematical properties which are relevant for informational analysis of genomes. Procedures to generate spectral segmentations of genomes are provided and are tested (under several values of length k for k-mers) on cases of real genomes, such as some human chromosomes and Saccharomyces cervisiae.

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“…The authors of [8] also proposed two encoding schemes of ℓ-repeat free strings; the first one uses a single redundancy symbol and supports ℓ = 2⌈log(n)⌉ + 2, while the second works for substrings of length ℓ = ⌈log n⌉ + ⌈2 log log n⌉ + 5 and its asymptotic rate approaches 1. Extensions of this problem to the setup where the ℓ-profiles are not received error-free were also studied recently [5], [9], [19], [28].…”

Section: Introductionmentioning

confidence: 99%

Multi-strand Reconstruction from Substrings

Yehezkeally,

Marcovich,

Yaakobi

2021

Preprint

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The problem of string reconstruction based on its substrings spectrum has received significant attention recently due to its applicability to DNA data storage and sequencing. In contrast to previous works, we consider in this paper a setup of this problem where multiple strings are reconstructed together. Given a multiset S of strings, all their substrings of some fixed length ℓ, defined as the ℓ-profile of S, are received and the goal is to reconstruct all strings in S. A multi-strand ℓ-reconstruction code is a set of multisets such that every element S can be reconstructed from its ℓ-profile. Given the number of strings k and their length n, we first find a lower bound on the value of ℓ necessary for existence of multi-strand ℓ-reconstruction codes with non-vanishing asymptotic rate. We then present two constructions of such codes and show that their rates approach 1 for values of ℓ that asymptotically behave like the lower bound.

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Reconstruction of Strings from their Substrings Spectrum

Cited by 10 publications

References 21 publications

A New Algebraic Approach for String Reconstruction from Substring Compositions

A New Algebraic Approach for String Reconstruction from Substring Compositions

Spectral concepts in genome informational analysis

Multi-strand Reconstruction from Substrings

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