English: The score matrix for an alignment between two nucleotide sequences:

Sequences    Best Alignments
---------    ----------------------
GCATGCG      GCATG-CG      GCA-TGCG      GCAT-GCG
GATTACA      G-ATTACA      G-ATTACA      G-ATTACA

The Needleman-Wunsch algorithm is a dynamic programming algorithm that finds the best possible alignments between two strings.

It takes three score parameters: match, mismatch, and gap.

Since it is a dynamic programming algorithm, the problem is solved by filling a two dimensional matrix. It requires O(nm) space and time where n and m are the lengths of the first and second string, respectively.

All best possible alignments are found by filling the matrix according to the defined score parameters.

Some optimizations of the algorithm avoid filling values distant from the diagonal, since those values are only needed for the poorest alignments.

Source code: https://gist.github.com/slowkow/508393