Subsequence

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. A subsequence is a more general term than a substring, which is a consecutive part of the original string.

Formally, suppose that X is a set and that (a_k)_{k ∈ K} is a sequence in X, where K = {1,2,3,...,n} if (a_k) is a finite sequence and K = N if (a_k) is an infinite sequence. Then, a subsequence of (a_k) is a sequence of the form

$(a_{n_{r}})$

where (n_r) is a strictly increasing sequence in the index set K.

Example

As an example,

$<B,C,D,B>$

is a subsequence of

$<A,C,B,D,E,G,C,E,D,B,G>$,

with corresponding index sequence <3,7,9,10>.

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.

See also

• subsequential limits
• limsup
• liminf
• longest common subsequence problem

subsequence at PlanetMath.