In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence is a subsequence of obtained after removal of elements C, E, and F. The relation of one sequence being the subsequence of another is a preorder. Subsequences can contain consecutive elements which were not consecutive in the original sequence. A subsequence which consists of a consecutive run of elements from the original sequence, such as from is a substring. The substring is a refinement of the subsequence. The list of all subsequences for the word "apple" would be "a", "ap", "al", "ae", "app", "apl", "ape", "ale", "appl", "appe", "aple", "apple", "p", "pp", "pl", "pe", "ppl", "ppe", "ple", "pple", "l", "le", "e", "" (empty string).

Common subsequence

Given two sequences X and Y, a sequence Z is said to be a common subsequence of X and Y, if Z is a subsequence of both X and Y. For example, if then Z is said to be a common subsequence of X and Y. This would be the longest common subsequence, since Z only has length 3, and the common subsequence has length 4. The longest common subsequence of X and Y is

Applications

Subsequences have applications to computer science, especially in the discipline of bioinformatics, where computers are used to compare, analyze, and store DNA, RNA, and protein sequences. Take two sequences of DNA containing 37 elements, say: The longest common subsequence of sequences 1 and 2 is: This can be illustrated by highlighting the 27 elements of the longest common subsequence into the initial sequences: Another way to show this is to align the two sequences, that is, to position elements of the longest common subsequence in a same column (indicated by the vertical bar) and to introduce a special character (here, a dash) for padding of arisen empty subsequences: Subsequences are used to determine how similar the two strands of DNA are, using the DNA bases: adenine, guanine, cytosine and thymine.

Theorems