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Common subsequence

Given two sequences X {\displaystyle X} and Y , {\displaystyle Y,} a sequence Z {\displaystyle Z} is said to be a common subsequence of X {\displaystyle X} and Y , {\displaystyle Y,} if Z {\displaystyle Z} is a subsequence of both X {\displaystyle X} and Y . {\displaystyle Y.} For example, if X = ⟨ A , C , B , D , E , G , C , E , D , B , G ⟩  and {\displaystyle X=\langle A,C,B,D,E,G,C,E,D,B,G\rangle \qquad {\text{ and}}} Y = ⟨ B , E , G , J , C , F , E , K , B ⟩  and {\displaystyle Y=\langle B,E,G,J,C,F,E,K,B\rangle \qquad {\text{ and}}} Z = ⟨ B , E , E ⟩ . {\displaystyle Z=\langle B,E,E\rangle .} then Z {\displaystyle Z} is said to be a common subsequence of X {\displaystyle X} and Y . {\displaystyle Y.}

This would not be the longest common subsequence, since Z {\displaystyle Z} only has length 3, and the common subsequence ⟨ B , E , E , B ⟩ {\displaystyle \langle B,E,E,B\rangle } has length 4. The longest common subsequence of X {\displaystyle X} and Y {\displaystyle Y} is ⟨ B , E , G , C , E , B ⟩ . {\displaystyle \langle B,E,G,C,E,B\rangle .}

Applications

Subsequences have applications to computer science,¹ especially in the discipline of bioinformatics, where computers are used to compare, analyze, and store DNA, RNA, and protein sequences.

Take two sequences of DNA containing 37 elements, say:

SEQ1 = ACGGTGTCGTGCTATGCTGATGCTGACTTATATGCTA SEQ2 = CGTTCGGCTATCGTACGTTCTATTCTATGATTTCTAA

The longest common subsequence of sequences 1 and 2 is:

LCS(SEQ1,SEQ2) = CGTTCGGCTATGCTTCTACTTATTCTA

This can be illustrated by highlighting the 27 elements of the longest common subsequence into the initial sequences:

SEQ1 = ACGGTGTCGTGCTATGCTGATGCTGACTTATATGCTA SEQ2 = CGTTCGGCTATCGTACGTTCTATTCTATGATTTCTAA

Another way to show this is to align the two sequences, that is, to position elements of the longest common subsequence in a same column (indicated by the vertical bar) and to introduce a special character (here, a dash) for padding of arisen empty subsequences:

SEQ1 = ACGGTGTCGTGCTAT-G--C-TGATGCTGA--CT-T-ATATG-CTA-         | || ||| ||||| |  | |  | || |  || | || |  ||| SEQ2 = -C-GT-TCG-GCTATCGTACGT--T-CT-ATTCTATGAT-T-TCTAA

Subsequences are used to determine how similar the two strands of DNA are, using the DNA bases: adenine, guanine, cytosine and thymine.

Theorems

• Every infinite sequence of real numbers has an infinite monotone subsequence. (This is a lemma used in the proof of the Bolzano–Weierstrass theorem.)
• Every infinite bounded sequence in R n {\displaystyle \mathbb {R} ^{n}} has a convergent subsequence. (This is the Bolzano–Weierstrass theorem.)
• For all integers r {\displaystyle r} and s , {\displaystyle s,} every finite sequence of length at least ( r − 1 ) ( s − 1 ) + 1 {\displaystyle (r-1)(s-1)+1} contains a monotonically increasing subsequence of length  r {\displaystyle r} or a monotonically decreasing subsequence of length  s {\displaystyle s} . (This is the Erdős–Szekeres theorem.)
• A metric space ( X , d ) {\displaystyle (X,d)} is compact if every sequence in X {\displaystyle X} has a convergent subsequence whose limit is in X {\displaystyle X} .

See also

• Subsequential limit – The limit of some subsequence
• Limit superior and limit inferior – Bounds of a sequencePages displaying short descriptions of redirect targets
• Longest increasing subsequence problem – Computer science problemPages displaying short descriptions of redirect targets

Notes

This article incorporates material from subsequence on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

References

1. In computer science, string is often used as a synonym for sequence, but it is important to note that substring and subsequence are not synonyms. Substrings are consecutive parts of a string, while subsequences need not be. This means that a substring of a string is always a subsequence of the string, but a subsequence of a string is not always a substring of the string, see: Gusfield, Dan (1999) [1997]. Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology. USA: Cambridge University Press. p. 4. ISBN 0-521-58519-8. 0-521-58519-8 ↩