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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 is a subsequence of obtained after removal of elements and The relation of one sequence being the subsequence of another is a preorder.

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 from 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).

Common subsequence[edit]

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

then is said to be a common subsequence of and

This would not be the longest common subsequence, since only has length 3, and the common subsequence has length 4. The longest common subsequence of and is


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:


The longest common subsequence of sequences 1 and 2 is:


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


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:

        | || ||| ||||| |  | |  | || |  || | || |  |||

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


  • 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 has a convergent subsequence (This is the Bolzano–Weierstrass theorem).
  • For all integers and every finite sequence of length at least contains a monotonically increasing subsequence of length  or a monotonically decreasing subsequence of length  (This is the Erdős–Szekeres theorem).
  • A metric space is compact if every sequence in has a convergent subsequence whose limit is in .

See also[edit]


  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.