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 is a subsequence of . They should not be confused with substring which is for above string and which is a refinement of subsequence. The relation of one sequence being the subsequence of another is a preorder.
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
then a common subsequence of X and Y could be
This would not be the longest common subsequence, since G only has length 3, and the common subsequence has length 4. The longest common subsequence of X and Y is .
Take two strands of DNA, say:
- ORG1 = ACGGTGTCGTGCTATGCTGATGCTGACTTATATGCTA
- ORG2 = CGTTCGGCTATCGTACGTTCTATTCTATGATTTCTAA.
- 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).
- Subsequential limit - the limit of some subsequence.
- Limit superior and limit inferior
- Longest increasing subsequence problem
- 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) . Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology. USA: Cambridge University Press. p. 4. ISBN 0-521-58519-8.