Isometric Words and Edit Distance: Main Notions and New Variations

Abstract

Isometric words combine the notion of edit distance together with properties of words not appearing as factors in other words. An edit distance is a metric between words that quantifies how two words differ by counting the number of edit operations needed to transform one word into the other one. A word f is said isometric with respect to an edit distance if, for any pair of f-free words u and v, there exists a transformation of minimal length from u into v via the related edit operations such that all the intermediate words are also f-free. The adjective “isometric” comes from the fact that, if the Hamming distance is considered (i.e., only replacement operations are used), then isometric words are connected with the definitions of isometric subgraphs of hypercubes. We discuss known results and some interesting generalizations and open problems.