Elastic-Degenerate String Matching with 1 Error or Mismatch

An elastic-degenerate (ED) string is a sequence of n finite sets of strings of total length N , introduced to represent a set of related DNA sequences, also known as a pangenome . The ED string matching (EDSM) problem consists in reporting all occurrences of a pattern of length m in an ED text. The...

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Published in:	Theory of computing systems Vol. 68; no. 5; pp. 1442 - 1467
Main Authors:	Bernardini, Giulia, Gabory, Esteban, Pissis, Solon P., Stougie, Leen, Sweering, Michelle, Zuba, Wiktor
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.10.2024 Springer Nature B.V Springer Verlag
Subjects:	Algorithms Bioinformatics Combinatorial analysis Computational geometry Computer Science Errors Gene sequencing Pattern matching Rectangles String matching Theory of Computation String algorithms Elastic-degenerate strings Edit distance Hamming distance Approximate string matching
ISSN:	1432-4350, 1433-0490
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Abstract	An elastic-degenerate (ED) string is a sequence of n finite sets of strings of total length N , introduced to represent a set of related DNA sequences, also known as a pangenome . The ED string matching (EDSM) problem consists in reporting all occurrences of a pattern of length m in an ED text. The EDSM problem has recently received some attention by the combinatorial pattern matching community, culminating in an O ~ ( n m ω - 1 ) + O ( N ) -time algorithm [Bernardini et al., SIAM J. Comput. 2022], where ω denotes the matrix multiplication exponent and the O ~ ( · ) notation suppresses polylog factors. In the k -EDSM problem, the approximate version of EDSM, we are asked to report all pattern occurrences with at most k errors. k -EDSM can be solved in O ( k 2 m G + k N ) time, under edit distance, or O ( k m G + k N ) time, under Hamming distance, where G denotes the total number of strings in the ED text [Bernardini et al., Theor. Comput. Sci. 2020]. Unfortunately, G is only bounded by N , and so even for k = 1 , the existing algorithms run in Ω ( m N ) time in the worst case. In this paper we make progress in this direction. We show that 1-EDSM can be solved in O ( ( n m 2 + N ) log m ) or O ( n m 3 + N ) time under edit distance. For the decision version of the problem, we present a faster O ( n m 2 log m + N log log m ) -time algorithm. We also show that 1-EDSM can be solved in O ( n m 2 + N log m ) time under Hamming distance. Our algorithms for edit distance rely on non-trivial reductions from 1-EDSM to special instances of classic computational geometry problems (2d rectangle stabbing or 2d range emptiness), which we show how to solve efficiently. In order to obtain an even faster algorithm for Hamming distance, we rely on employing and adapting the k -errata trees for indexing with errors [Cole et al., STOC 2004]. This is an extended version of a paper presented at LATIN 2022.
AbstractList	An elastic-degenerate (ED) string is a sequence of n finite sets of strings of total length N , introduced to represent a set of related DNA sequences, also known as a pangenome. The ED string matching (EDSM) problem consists in reporting all occurrences of a pattern of length m in an ED text. The EDSM problem has recently received some attention by the combinatorial pattern matching community, culminating in an Õ(nm ω-1 ) + O(N )-time algorithm [Bernardini et al., SIAM J. Comput. 2022], where ω denotes the matrix multiplication exponent and the Õ(•) notation suppresses polylog factors. In the k-EDSM problem, the approximate version of EDSM, we are asked to report all pattern occurrences with at most k errors. k-EDSM can be solved in O(k 2 mG + k N ) time, under edit distance, or O(kmG + k N ) time, under Hamming distance, where G denotes the total number of strings in the ED text [Bernardini et al., Theor. Comput. Sci. 2020]. Unfortunately, G is only bounded by N , and so even for k = 1, the existing algorithms run in Ω(m N ) time in the worst case. In this paper we make progress in this direction. We show that 1-EDSM can be solved in O((nm 2 + N ) log m) or O(nm 3 + N ) time under edit distance. For the decision version of the problem, we present a faster O(nm 2 √ log m + N log log m)-time algorithm. We also show that 1-EDSM can be solved in O(nm 2 + N log m) time under Hamming distance. Our algorithms for edit distance rely on non-trivial reductions from 1-EDSM to special instances of classic computational geometry problems (2d rectangle stabbing or 2d range emptiness), which we show how to solve efficiently. In order to obtain an even faster algorithm for Hamming distance, we rely on employing and adapting the k-errata trees for indexing with errors [Cole et al., STOC 2004]. An elastic-degenerate (ED) string is a sequence of n finite sets of strings of total length N, introduced to represent a set of related DNA sequences, also known as a pangenome. The ED string matching (EDSM) problem consists in reporting all occurrences of a pattern of length m in an ED text. The EDSM problem has recently received some attention by the combinatorial pattern matching community, culminating in an O~(nmω-1)+O(N)-time algorithm [Bernardini et al., SIAM J. Comput. 2022], where ω denotes the matrix multiplication exponent and the O~(·) notation suppresses polylog factors. In the k-EDSM problem, the approximate version of EDSM, we are asked to report all pattern occurrences with at most k errors. k-EDSM can be solved in O(k2mG+kN) time, under edit distance, or O(kmG+kN) time, under Hamming distance, where G denotes the total number of strings in the ED text [Bernardini et al., Theor. Comput. Sci. 2020]. Unfortunately, G is only bounded by N, and so even for k=1, the existing algorithms run in Ω(mN) time in the worst case. In this paper we make progress in this direction. We show that 1-EDSM can be solved in O((nm2+N)logm) or O(nm3+N) time under edit distance. For the decision version of the problem, we present a faster O(nm2logm+Nloglogm)-time algorithm. We also show that 1-EDSM can be solved in O(nm2+Nlogm) time under Hamming distance. Our algorithms for edit distance rely on non-trivial reductions from 1-EDSM to special instances of classic computational geometry problems (2d rectangle stabbing or 2d range emptiness), which we show how to solve efficiently. In order to obtain an even faster algorithm for Hamming distance, we rely on employing and adapting the k-errata trees for indexing with errors [Cole et al., STOC 2004]. This is an extended version of a paper presented at LATIN 2022. An elastic-degenerate (ED) string is a sequence of n finite sets of strings of total length N , introduced to represent a set of related DNA sequences, also known as a pangenome . The ED string matching (EDSM) problem consists in reporting all occurrences of a pattern of length m in an ED text. The EDSM problem has recently received some attention by the combinatorial pattern matching community, culminating in an O ~ ( n m ω - 1 ) + O ( N ) -time algorithm [Bernardini et al., SIAM J. Comput. 2022], where ω denotes the matrix multiplication exponent and the O ~ ( · ) notation suppresses polylog factors. In the k -EDSM problem, the approximate version of EDSM, we are asked to report all pattern occurrences with at most k errors. k -EDSM can be solved in O ( k 2 m G + k N ) time, under edit distance, or O ( k m G + k N ) time, under Hamming distance, where G denotes the total number of strings in the ED text [Bernardini et al., Theor. Comput. Sci. 2020]. Unfortunately, G is only bounded by N , and so even for k = 1 , the existing algorithms run in Ω ( m N ) time in the worst case. In this paper we make progress in this direction. We show that 1-EDSM can be solved in O ( ( n m 2 + N ) log m ) or O ( n m 3 + N ) time under edit distance. For the decision version of the problem, we present a faster O ( n m 2 log m + N log log m ) -time algorithm. We also show that 1-EDSM can be solved in O ( n m 2 + N log m ) time under Hamming distance. Our algorithms for edit distance rely on non-trivial reductions from 1-EDSM to special instances of classic computational geometry problems (2d rectangle stabbing or 2d range emptiness), which we show how to solve efficiently. In order to obtain an even faster algorithm for Hamming distance, we rely on employing and adapting the k -errata trees for indexing with errors [Cole et al., STOC 2004]. This is an extended version of a paper presented at LATIN 2022. An elastic-degenerate (ED) string is a sequence of n finite sets of strings of total length N , introduced to represent a set of related DNA sequences, also known as a pangenome . The ED string matching (EDSM) problem consists in reporting all occurrences of a pattern of length m in an ED text. The EDSM problem has recently received some attention by the combinatorial pattern matching community, culminating in an $$\mathcal {\tilde{O}}(nm^{\omega -1})+\mathcal {O}(N)$$ O ~ ( n m ω - 1 ) + O ( N ) -time algorithm [Bernardini et al., SIAM J. Comput. 2022], where $$\omega $$ ω denotes the matrix multiplication exponent and the $$\mathcal {\tilde{O}}(\cdot )$$ O ~ ( · ) notation suppresses polylog factors. In the k -EDSM problem, the approximate version of EDSM, we are asked to report all pattern occurrences with at most k errors. k -EDSM can be solved in $$\mathcal {O}(k^2mG+kN)$$ O ( k 2 m G + k N ) time, under edit distance, or $$\mathcal {O}(kmG+kN)$$ O ( k m G + k N ) time, under Hamming distance, where G denotes the total number of strings in the ED text [Bernardini et al., Theor. Comput. Sci. 2020]. Unfortunately, G is only bounded by N , and so even for $$k=1$$ k = 1 , the existing algorithms run in $$\varOmega (mN)$$ Ω ( m N ) time in the worst case. In this paper we make progress in this direction. We show that 1-EDSM can be solved in $$\mathcal {O}((nm^2 + N)\log m)$$ O ( ( n m 2 + N ) log m ) or $$\mathcal {O}(nm^3 + N)$$ O ( n m 3 + N ) time under edit distance. For the decision version of the problem, we present a faster $$\mathcal {O}(nm^2\sqrt{\log m} + N\log \log m)$$ O ( n m 2 log m + N log log m ) -time algorithm. We also show that 1-EDSM can be solved in $$\mathcal {O}(nm^2 + N\log m)$$ O ( n m 2 + N log m ) time under Hamming distance. Our algorithms for edit distance rely on non-trivial reductions from 1-EDSM to special instances of classic computational geometry problems (2d rectangle stabbing or 2d range emptiness), which we show how to solve efficiently. In order to obtain an even faster algorithm for Hamming distance, we rely on employing and adapting the k -errata trees for indexing with errors [Cole et al., STOC 2004]. This is an extended version of a paper presented at LATIN 2022.
Author	Sweering, Michelle Pissis, Solon P. Gabory, Esteban Zuba, Wiktor Stougie, Leen Bernardini, Giulia
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Keywords	String algorithms Elastic-degenerate strings Edit distance Hamming distance Approximate string matching
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Snippet	An elastic-degenerate (ED) string is a sequence of n finite sets of strings of total length N , introduced to represent a set of related DNA sequences, also... An elastic-degenerate (ED) string is a sequence of n finite sets of strings of total length N, introduced to represent a set of related DNA sequences, also... An elastic-degenerate (ED) string is a sequence of n finite sets of strings of total length N , introduced to represent a set of related DNA sequences, also...
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SubjectTerms	Algorithms Bioinformatics Combinatorial analysis Computational geometry Computer Science Errors Gene sequencing Pattern matching Rectangles String matching Theory of Computation
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Title	Elastic-Degenerate String Matching with 1 Error or Mismatch
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