Subsequence

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:

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.

Theorems

• 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 ${\displaystyle X}$.