Editing Dynamic programming (section)

=== Sequence alignment ===
In [[genetics]], [[sequence alignment]] is an important application where dynamic programming is essential.<ref name="Eddy"/>  Typically, the problem consists of transforming one sequence into another using edit operations that replace, insert, or remove an element.  Each operation has an associated cost, and the goal is to find the [[edit distance|sequence of edits with the lowest total cost]].

The problem can be stated naturally as a recursion, a sequence A is optimally edited into a sequence B by either:

# inserting the first character of B, and performing an optimal alignment of A and the tail of B
# deleting the first character of A, and performing the optimal alignment of the tail of A and B
# replacing the first character of A with the first character of B, and performing optimal alignments of the tails of A and B.

The partial alignments can be tabulated in a matrix, where cell (i,j) contains the cost of the optimal alignment of A[1..i] to B[1..j].  The cost in cell (i,j) can be calculated by adding the cost of the relevant operations to the cost of its neighboring cells, and selecting the optimum.

Different variants exist, see [[Smith–Waterman algorithm]] and [[Needleman–Wunsch algorithm]].