Subsequence

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,[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:

SEQ₁ = ACGGTGTCGTGCTATGCTGATGCTGACTTATATGCTA

SEQ₂ = CGTTCGGCTATCGTACGTTCTATTCTATGATTTCTAA

The longest common subsequence of sequences 1 and 2 is:

LCS_(SEQ1,SEQ2) = CGTTCGGCTATGCTTCTACTTATTCTA

This can be illustrated by highlighting the 27 elements of the longest common subsequence into the initial sequences:

SEQ₁ = ACGGTGTCGTGCTATGCTGATGCTGACTTATATGCTA

SEQ₂ = CGTTCGGCTATCGTACGTTCTATTCTATGATTTCTAA

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:

SEQ₁ = ACGGTGTCGTGCTAT-G--C-TGATGCTGA--CT-T-ATATG-CTA-

        | || ||| ||||| |  | |  | || |  || | || |  |||

SEQ₂ = -C-GT-TCG-GCTATCGTACGT--T-CT-ATTCTATGAT-T-TCTAA

Subsequences are used to determine how similar the two strands of DNA are, using the DNA bases: adenine, guanine, cytosine and thymine.

• 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 infinite bounded sequence in ${\displaystyle \mathbb {R} ^{n}}$ has a convergent subsequence (This is the Bolzano–Weierstrass theorem).
• For all integers ${\displaystyle r}$ and ${\displaystyle s,}$ every finite sequence of length at least ${\displaystyle (r-1)(s-1)+1}$ contains a monotonically increasing subsequence of length ${\displaystyle r}$ or a monotonically decreasing subsequence of length ${\displaystyle s}$ (This is the Erdős–Szekeres theorem).
• A metric space ${\displaystyle (X,d)}$ is compact if every sequence in ${\displaystyle X}$ has a convergent subsequence whose limit is in ${\displaystyle X}$.

• Subsequential limit – The limit of some subsequence
• Limit superior and limit inferior – Bounds of a sequencePages displaying short descriptions of redirect targets
• Longest increasing subsequence problem – Computer science problemPages displaying short descriptions of redirect targets

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