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In mathematics, a subsequence is a sequence that can be derived from another 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 , , and . 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).
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 is said to be a common subsequence of X and Y.
This would not 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 .
Subsequences have applications to computer science,^{ [1] } 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, i.e., 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.
This article incorporates material from subsequence on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.
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