Classes of cost functions for string edit distance

TLDR: It is shown that cost functions having the same ratio of the sum of the insertion and deletion costs divided by the substitution cost yield the same minimum cost sequences of edit operations, which leads to a partitioning of the universe of cost functions into equivalence classes.

Abstract: Finding a sequence of edit operations that transforms one string of symbols into another with the minimum cost is a well-known problem. The minimum cost, or edit distance, is a widely used measure of the similarity of two strings. An important parameter of this problem is the cost function, which specifies the cost of each insertion, deletion, and substitution. We show that cost functions having the same ratio of the sum of the insertion and deletion costs divided by the substitution cost yield the same minimum cost sequences of edit operations. This leads to a partitioning of the universe of cost functions into equivalence classes. Also, we show the relationship between a particular set of cost functions and the longest common subsequence of the input strings.