Subsequence

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 ${\displaystyle \langle A,B,D\rangle }$ is a subsequence of ${\displaystyle \langle A,B,C,D,E,F\rangle }$ obtained after removal of elements ${\displaystyle C,}$ ${\displaystyle E,}$ and ${\displaystyle F.}$ The relation of one sequence being the subsequence of another is a partial order.

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 ${\displaystyle \langle B,C,D\rangle ,}$ from ${\displaystyle \langle A,B,C,D,E,F\rangle ,}$ 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 ${\displaystyle X}$ and ${\displaystyle Y,}$ a sequence ${\displaystyle Z}$ is said to be a common subsequence of ${\displaystyle X}$ and ${\displaystyle Y,}$ if ${\displaystyle Z}$ is a subsequence of both ${\displaystyle X}$ and ${\displaystyle Y.}$ For example, if $${\displaystyle X=\langle A,C,B,D,E,G,C,E,D,B,G\rangle \qquad {\text{ and}}}$$ $${\displaystyle Y=\langle B,E,G,J,C,F,E,K,B\rangle \qquad {\text{ and}}}$$ $${\displaystyle Z=\langle B,E,E\rangle .}$$ then ${\displaystyle Z}$ is said to be a common subsequence of ${\displaystyle X}$ and ${\displaystyle Y.}$

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

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: