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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  

Applications edit

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.

Theorems edit

  • 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

Notes 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.