Edit distance and comparison scores

The simplest form of sequence comparison is based on editing strings and counting the number of edits required to get from one sequence to the other. The formulation of string-edit distance de balances two different types of edits. The simplest is replacement of a single letter by another letter. To start with, we need a metric on the set of letters in the alphabet Σ for our sets of sequences. Let DΣ be a metric on the alphabet Σ. Then DΣ(ξ, η) measures the difference between the two letters ξ and η in Σ. Now we show how to extend this to a metric de on the set of all finite strings Σ . If two strings x and y differ only in the k-th position, then we set de(x, y) = DΣ(xk, yk). In general, when there are multiple replacements, string edit distance is based on just summing the effects. However, string-edit distance also allows a different kind of change as well: insertion and deletion. For example, we can define x k̂ to mean the string x with the k-th entry removed. It might be that x k̂ agrees perfectly with the string y, and so we assign d(x, y) = δ where δ is the deletion penalty. Similarly, insertions of characters are allowed to determine edit distance. Clearly, if y = x k̂ , then adding xk to y at the k-th position yields x. Again, the effect of multiple insertions/deletions is additive, and this allows strings of different lengths to be compared. The use of both replacements and instertion/deletions to determine edit distance means that an edit path from x to y is not unique. Edit distance is therefore defined by taking the minimum over all possible representations, as we define formally in (1.4). But this will not in general define a metric unless appropriate conditions on δ and DΣ are satisfied. These conditions can be defined by extending the alphabet Σ and metric DΣ to include a “gap” as a character, say “ ” (let Σ̃ denote the extended alphabet), and by assigning a distance De Σ(x, ) for each character x in the original alphabet. Theorem 9.4 of [1] says that de is a metric on strings of letters in Σ whenever De Σ is a metric on the extended alphabet.