subsequence

In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements. For example, ABD is a subsequence of ABCDEF.

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 where (n_r) 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 a 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 a subsequence of a string is not always a substring of the string.^[1]

See also

• subsequential limit
• limit superior and limit inferior
• longest common subsequence problem
• longest increasing subsequence problem
• Erdős–Szekeres theorem

References

1. ^ Gusfield, Dan (1999) [1997]. Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology. USA: Cambridge University Press. pp. 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.