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, the sequence $\langle A,B,D \rangle$ is a subsequence of $\langle A,B,C,D,E,F \rangle$. They should not be confused with substring which is $\langle A,B,C,D \rangle$ for above string and which is a refinement of subsequence.

Common subsequence

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

$X = \langle A,C,B,D,E,G,C,E,D,B,G \rangle$ and

$Y = \langle B,E,G,C,F,E,U,B,K \rangle$

then a common subsequence of X and Y could be

$G = \langle B,E,E \rangle.$

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

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
and

ORG₂ = CGTTCGGCTATCGTACGTTCTATTCTATGATTTCTAA.

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 bounded infinite sequence in Rⁿ has a convergent subsequence (This is the Bolzano–Weierstrass theorem).
• For every 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

This article incorporates material from subsequence on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.