Longest common substrings with k mismatches
The longest common substring with k-mismatches problem is to find, given two strings S1 and S2, a longest substring A1 of S1 and A2 of S2 such that the Hamming distance between A1 and A2 is ≤k. We introduce a practical O(nm) time and O(1) space solution for this problem, where n and m are the length...
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| Published in: | Information processing letters Vol. 115; no. 6-8; pp. 643 - 647 |
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| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
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Amsterdam
Elsevier B.V
01.06.2015
Elsevier Sequoia S.A |
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| ISSN: | 0020-0190, 1872-6119 |
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| Abstract | The longest common substring with k-mismatches problem is to find, given two strings S1 and S2, a longest substring A1 of S1 and A2 of S2 such that the Hamming distance between A1 and A2 is ≤k. We introduce a practical O(nm) time and O(1) space solution for this problem, where n and m are the lengths of S1 and S2, respectively. This algorithm can also be used to compute the matching statistics with k-mismatches of S1 and S2 in O(nm) time and O(m) space. Moreover, we also present a theoretical solution for the k=1 case which runs in O(nlogm) time, assuming m≤n, and uses O(m) space, improving over the existing O(nm) time and O(m) space bound of Babenko and Starikovskaya [1].
•Two new algorithms for the longest common substring with k mismatches problem.•A practical solution for arbitrary k which uses constant space.•A theoretical solution for one mismatch which runs in quasilinear time. |
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| AbstractList | The longest common substring with k-mismatches problem is to find, given two strings S1 and S2, a longest substring A1 of S1 and A2 of S2 such that the Hamming distance between A1 and A2 is ≤k. We introduce a practical O(nm) time and O(1) space solution for this problem, where n and m are the lengths of S1 and S2, respectively. This algorithm can also be used to compute the matching statistics with k-mismatches of S1 and S2 in O(nm) time and O(m) space. Moreover, we also present a theoretical solution for the k=1 case which runs in O(nlogm) time, assuming m≤n, and uses O(m) space, improving over the existing O(nm) time and O(m) space bound of Babenko and Starikovskaya [1].
•Two new algorithms for the longest common substring with k mismatches problem.•A practical solution for arbitrary k which uses constant space.•A theoretical solution for one mismatch which runs in quasilinear time. The longest common substring with k-mismatches problem is to find, given two strings S1 and S2, a longest substring A1 of S1 and A2 of S2 such that the Hamming distance between A1 and A2 is ... We introduce a practical O(nm) time and O(1) space solution for this problem, where n and m are the lengths of S1 and S2, respectively. This algorithm can also be used to compute the matching statistics with k-mismatches of S1 and S2 in O(nm) time and O(m) space. Moreover, we also present a theoretical solution for the k=1 case which runs in ... time, assuming ..., and uses O(m) space, improving over the existing O(nm) time and O(m) space bound of Babenko and Starikovskaya. (ProQuest: ... denotes formulae/symbols omitted.) |
| Author | Flouri, Tomas Giaquinta, Emanuele Ukkonen, Esko Kobert, Kassian |
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| Cites_doi | 10.1016/j.tcs.2006.06.029 10.1145/322123.322127 10.1017/CBO9780511574931 10.1016/0022-2836(81)90087-5 10.1134/S0032946011010030 10.1007/BF01185431 10.1093/bioinformatics/btu331 |
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| Keywords | String algorithms Combinatorial problems Longest common substring Hamming distance |
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| References_xml | – volume: 12 start-page: 327 year: 1994 end-page: 344 ident: br0060 article-title: Sublinear approximate string matching and biological applications publication-title: Algorithmica – start-page: 223 year: 2013 end-page: 234 ident: br0030 article-title: Time-space trade-offs for the longest common substring problem publication-title: CPM – volume: 362 start-page: 248 year: 2006 end-page: 254 ident: br0080 article-title: Longest repeats with a block of publication-title: Theor. Comput. Sci. – start-page: 220 year: 1986 end-page: 230 ident: br0100 article-title: Introducing efficient parallelism into approximate string matching and a new serial algorithm publication-title: STOC – volume: 47 start-page: 28 year: 2011 end-page: 33 ident: br0010 article-title: Computing the longest common substring with one mismatch publication-title: Probl. Inf. Transm. – volume: 30 start-page: 2000 year: 2014 end-page: 2008 ident: br0070 article-title: kmacs: the publication-title: Bioinformatics – volume: 147 start-page: 195 year: 1981 end-page: 197 ident: br0050 article-title: Identification of common molecular subsequences publication-title: J. Mol. Biol. – start-page: 605 year: 2014 end-page: 617 ident: br0040 article-title: Sublinear space algorithms for the longest common substring problem publication-title: ESA – year: 1997 ident: br0020 article-title: Algorithms on Strings, Trees, and Sequences publication-title: Computer Science and Computational Biology – volume: 26 start-page: 211 year: 1979 end-page: 226 ident: br0110 article-title: A fast merging algorithm publication-title: J. ACM – start-page: 88 year: 2000 end-page: 94 ident: br0090 article-title: The LCA problem revisited publication-title: LATIN – volume: 362 start-page: 248 issue: 1–3 year: 2006 ident: 10.1016/j.ipl.2015.03.006_br0080 article-title: Longest repeats with a block of k don't cares publication-title: Theor. Comput. Sci. doi: 10.1016/j.tcs.2006.06.029 – start-page: 220 year: 1986 ident: 10.1016/j.ipl.2015.03.006_br0100 article-title: Introducing efficient parallelism into approximate string matching and a new serial algorithm – volume: 26 start-page: 211 issue: 2 year: 1979 ident: 10.1016/j.ipl.2015.03.006_br0110 article-title: A fast merging algorithm publication-title: J. ACM doi: 10.1145/322123.322127 – year: 1997 ident: 10.1016/j.ipl.2015.03.006_br0020 article-title: Algorithms on Strings, Trees, and Sequences doi: 10.1017/CBO9780511574931 – start-page: 88 year: 2000 ident: 10.1016/j.ipl.2015.03.006_br0090 article-title: The LCA problem revisited – volume: 147 start-page: 195 issue: 1 year: 1981 ident: 10.1016/j.ipl.2015.03.006_br0050 article-title: Identification of common molecular subsequences publication-title: J. Mol. Biol. doi: 10.1016/0022-2836(81)90087-5 – volume: 47 start-page: 28 issue: 1 year: 2011 ident: 10.1016/j.ipl.2015.03.006_br0010 article-title: Computing the longest common substring with one mismatch publication-title: Probl. Inf. Transm. doi: 10.1134/S0032946011010030 – volume: 12 start-page: 327 issue: 4/5 year: 1994 ident: 10.1016/j.ipl.2015.03.006_br0060 article-title: Sublinear approximate string matching and biological applications publication-title: Algorithmica doi: 10.1007/BF01185431 – volume: 30 start-page: 2000 issue: 14 year: 2014 ident: 10.1016/j.ipl.2015.03.006_br0070 article-title: kmacs: the k-mismatch average common substring approach to alignment-free sequence comparison publication-title: Bioinformatics doi: 10.1093/bioinformatics/btu331 – start-page: 223 year: 2013 ident: 10.1016/j.ipl.2015.03.006_br0030 article-title: Time-space trade-offs for the longest common substring problem – start-page: 605 year: 2014 ident: 10.1016/j.ipl.2015.03.006_br0040 article-title: Sublinear space algorithms for the longest common substring problem |
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| SubjectTerms | Algorithms Combinatorial problems Hamming distance Longest common substring Mathematical problems String algorithms Studies Vector space |
| Title | Longest common substrings with k mismatches |
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