Definitions

# Subsequence

[suhb-si-kwuhns]
In mathematics, a subsequence of some sequence is a new sequence which is formed from the original sequence by deleting some of the elements without disturbing the relative positions of the remaining elements.

Formally, suppose that X is a set and that (ak)kK is a sequence in X, where K = {1,2,3,...,n} if (ak) is a finite sequence and K = N if (ak) is an infinite sequence. Then, a subsequence of (ak) is a sequence of the form $\left(a_\left\{n_r\right\}\right)$ where (nr) is a strictly increasing sequence in the index set K.

## Example

As an example,
$< B,C,D,G >$
is a subsequence of
$< A,C,B,D,E,G,C,E,D,B,G >$,
with corresponding index sequence <3,7,9,11>.

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

$X = < A,C,B,D,E,G,C,E,D,B,G >$ and
$Y = < B,E,G,C,F,E,U,B,K >$
then common subsequence of X and Y could be
$G = < B,E,E >$

This would not be the longest common subsequence, since G only has length 3, and the common subsequence < B,E,E,B > has length 4. The longest common subsequence of X and Y is < B,E,G,C,E,B >.

## Applications

Subsequences have applications to computer science, especially in the discipline of Bioinformatics, where computers are used to compare, analyze, and store DNA strands.

Take two strands of DNA, say :

ORG1 = ACGGTGTCGTGCTATGCTGATGCTGACTTATATGCTA
ORG2 = CGTTCGGCTATCGTACGTTCTATTCTATGATTTCTAA

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

## Substring vs. subsequence

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 the subsequence of a string is not always a substring of the string.