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In mathematics, a **subsequence** is a sequence that can be derived from another 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*", "".

## 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

- and

- and

then 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 has length 4. The longest common subsequence of *X* and *Y* is .

## 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:

`SEQ`_{1}= ACGGTGTCGTGCTATGCTGATGCTGACTTATATGCTA`SEQ`_{2}= 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:

`SEQ`_{1}= A**CG**G**T**G**TCG**T**GCTATGCT**GA**T**G**CT**G**ACTTAT**A**T**G**CTA**`SEQ`_{2}=**CGTTCGGCTAT**C**G**TA**C**G**TTCTA**TT**CT**A**T**G**ATT**T**CTA**A

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:

`SEQ`_{1}= ACGGTGTCGTGCTAT-G--C-TGATGCTGA--CT-T-ATATG-CTA-`| || ||| ||||| | | | | || | || | || | |||``SEQ`_{2}= -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}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).

## See also

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

## Notes

**^**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.*