Point Mutations Include Insertions and Deletions
In “Counting Point Mutations”, we saw that Hamming distance gave us a preliminary notion of the evolutionary distance between two DNA strings by counting the minimum number of single nucleotide substitutions that could have occurred on the evolutionary path between the two strands.
However, in practice, homologous strands of DNA or protein are rarely the same length because point mutations also include the insertion or deletion of a single nucleotide (and single amino acids can be inserted or deleted from peptides). Thus, we need to incorporate these insertions and deletions into the calculation of the minimum number of point mutations between two strings. One of the simplest models charges a unit "cost" to any single-symbol insertion/deletion, then (in keeping with parsimony) requests the minimum cost over all transformations of one genetic string into another by point substitutions, insertions, and deletions.
Given two strings
The latter two operations incorporate the case in which a contiguous interval is inserted into or deleted from a string;
such an interval is called a gap.
For the purposes of this problem, the insertion or deletion of a gap of length
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