Subsequence

In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements. For example, the sequence $\langle A,B,D\rangle$ is a subsequence of $\langle A,B,C,D,E,F\rangle$ obtained after removal of elements $C$, $E$, and $F$. The relation of one sequence being the subsequence of another is a preorder.

The subsequence should not be confused with substring $\langle A,B,C,D\rangle$ which can be derived from the above string $\langle A,B,C,D,E,F\rangle$ by deleting substring ${\displaystyle \langle E,F\rangle }$. The substring is a refinement of the subsequence.

Common subsequence

Given two sequences X and Y, a sequence Z is said to be a common subsequence of X and Y, if Z is a subsequence of both X and Y. For example, if

$X=\langle A,C,B,D,E,G,C,E,D,B,G\rangle$ and

$Y=\langle B,E,G,C,F,E,U,B,K\rangle$

then a common subsequence of X and Y could be

${\displaystyle Z=\langle B,E,E\rangle .}$

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

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, i.e., 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) in one sequence when two elements in the same column differ:

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 bounded infinite sequence in Rⁿ has a convergent subsequence (This is the Bolzano–Weierstrass theorem).
• For every integers r and s, every finite sequence of length at least (r − 1)(s − 1) + 1 contains a monotonically increasing subsequence of length r or a monotonically decreasing subsequence of length s (This is the Erdős–Szekeres theorem).

See also

• Subsequential limit - the limit of some subsequence.
• Limit superior and limit inferior
• Longest increasing subsequence problem

Notes

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.

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