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Subsequence

Mathematical binary relation

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Source updated May 19, 2026

In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no 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 partial order.

Subsequences can contain consecutive elements which were not consecutive in the original sequence. A subsequence which consists of a consecutive run of elements from the original sequence, such as $\langle B,C,D\rangle ,$ from $\langle A,B,C,D,E,F\rangle ,$ is a substring. The substring is a refinement of the subsequence.

The list of all subsequences for the word "apple" would be "a", "ap", "al", "ae", "app", "apl", "ape", "ale", "appl", "appe", "aple", "apple", "p", "pp", "pl", "pe", "ppl", "ppe", "ple", "pple", "l", "le", "e", "" (empty string).

01Common 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 $X=\langle A,C,B,D,E,G,C,E,D,B,G\rangle \qquad {\text{ and}}$ $Y=\langle B,E,G,J,C,F,E,K,B\rangle \qquad {\text{ and}}$ $Z=\langle B,E,E\rangle .$ then $Z$ is said to be a common subsequence of $X$ and $Y.$

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 $\langle B,E,G,C,E,B\rangle .$

02Applications

Subsequences have applications to computer science, 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.

03Theorems

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 $\mathbb {R} ^{n}$ has a convergent subsequence. (This is the Bolzano-Weierstrass theorem.)
For all 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.)
A metric space $(X,d)$ is compact if every sequence in $X$ has a convergent subsequence whose limit is in $X$ .

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Sources and credits

This article is adapted from the Wikipedia article “Subsequence”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Subsequence

01Common subsequence

02Applications

03Theorems

Sources and credits

Related topics

Subsequential limit

Limit inferior and limit superior

Longest increasing subsequence