Unique Reconstruction of Coded Strings From Multiset Substring Spectra

Gabrys, Ryan; Milenković, Olgica

doi:10.1109/tit.2019.2935973

Cited by 32 publications

(16 citation statements)

References 39 publications

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“…Therefore, a lower bound for D 3 (n) is We estimate the terms of (10). Note that if t = 0 then s ′′ < 0, then we can consider…”

Section: Lower Bounds On D 3 (N)mentioning

confidence: 99%

On the number of distinct k-decks: Enumeration and bounds

Chrisnata¹,

Kiah²,

Karingula³

et al. 2023

AMC

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The <inline-formula><tex-math id="M2">\begin{document}$ k $\end{document}</tex-math></inline-formula>-deck of a sequence is defined as the multiset of all its subsequences of length <inline-formula><tex-math id="M3">\begin{document}$ k $\end{document}</tex-math></inline-formula>. Let <inline-formula><tex-math id="M4">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> denote the number of distinct <inline-formula><tex-math id="M5">\begin{document}$ k $\end{document}</tex-math></inline-formula>-decks for binary sequences of length <inline-formula><tex-math id="M6">\begin{document}$ n $\end{document}</tex-math></inline-formula>. For binary alphabet, we determine the exact value of <inline-formula><tex-math id="M7">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> for small values of <inline-formula><tex-math id="M8">\begin{document}$ k $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ n $\end{document}</tex-math></inline-formula>, and provide asymptotic estimates of <inline-formula><tex-math id="M10">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M11">\begin{document}$ k $\end{document}</tex-math></inline-formula> is fixed.Specifically, for fixed <inline-formula><tex-math id="M12">\begin{document}$ k $\end{document}</tex-math></inline-formula>, we introduce a trellis-based method to compute <inline-formula><tex-math id="M13">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> in time polynomial in <inline-formula><tex-math id="M14">\begin{document}$ n $\end{document}</tex-math></inline-formula>. We then compute <inline-formula><tex-math id="M15">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M16">\begin{document}$ k \in \{3,4,5,6\} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M17">\begin{document}$ k \leqslant n \leqslant 30 $\end{document}</tex-math></inline-formula>. We also improve the asymptotic upper bound on <inline-formula><tex-math id="M18">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula>, and provide a lower bound thereupon. In particular, for binary alphabet, we show that <inline-formula><tex-math id="M19">\begin{document}$ D_k(n) = O\bigl(n^{(k-1)2^{k-1}+1}\bigr) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M20">\begin{document}$ D_k(n) = \Omega(n^k) $\end{document}</tex-math></inline-formula>. For <inline-formula><tex-math id="M21">\begin{document}$ k = 3 $\end{document}</tex-math></inline-formula>, we moreover show that <inline-formula><tex-math id="M22">\begin{document}$ D_3(n) = \Omega(n^6) $\end{document}</tex-math></inline-formula> while the upper bound on <inline-formula><tex-math id="M23">\begin{document}$ D_3(n) $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M24">\begin{document}$ O(n^9) $\end{document}</tex-math></inline-formula>.

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“…Therefore, a lower bound for D 3 (n) is We estimate the terms of (10). Note that if t = 0 then s ′′ < 0, then we can consider…”

Section: Lower Bounds On D 3 (N)mentioning

confidence: 99%

On the number of distinct k-decks: Enumeration and bounds

Chrisnata¹,

Kiah²,

Karingula³

et al. 2023

AMC

View full text Add to dashboard Cite

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“…Since all the prefixes in G pw,hq have weight w, it is straightforward to verify that the set of codewords from (12) and the set in (13) have the same prefix composition multisets.…”

Section: And Thatmentioning

confidence: 99%

Reconstruction of Sets of Strings from Prefix/Suffix Compositions

Gabrys¹,

Pattabiraman²,

Milenković³

2021

Preprint

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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 B h 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 B h 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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“…The problem of reconstructing strings based on evidence sets of the form of subsequences, substrings or weights of substrings has received significant attention from the theoretical computer science, bioinformatics, and information theory communities alike [1], [3], [4], [8], [10], [11], [13], [15]. One special instance of this class of problems is the k-deck problem [4], [6], [7], [9], [10], [14], of interest due to its connection to trace reconstruction [3], [5] and its applications in DNA-based data storage [16].…”

Section: Introductionmentioning

confidence: 99%

The Gapped $k$-Deck Problem

Golm¹,

Nahvi²,

Gabrys³

et al. 2022

Preprint

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The k-deck problem is concerned with finding the smallest value Spkq of a positive integer n such that there exist at least two strings of length n that share the same k-deck, i.e., the same 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 possible value of n, G s pkq, such that there exist at least two distinct strings of length n that cannot be distinguished based on a "gapped" set of ksubsequences. The gap constraint requires the elements in the subsequences to be at least s positions apart in 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 pkq. Second, we further improve this upper bound in a nonconstructive manner using the approach by Dudik and Schulman [6]. Third, we comment on the general case s ě 2 and present a number of open problems.

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Unique Reconstruction of Coded Strings From Multiset Substring Spectra

Cited by 32 publications

References 39 publications

On the number of distinct <i>k</i>-decks: Enumeration and bounds

On the number of distinct <i>k</i>-decks: Enumeration and bounds

Reconstruction of Sets of Strings from Prefix/Suffix Compositions

The Gapped $k$-Deck Problem

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