Metric Indexes for Approximate String Matching in a Dictionary

Abstract

We consider the problem of finding all approximate occurrences of a given string q, with at most k differences, in a finite database or dictionary of strings. The strings can be e.g. natural language words, such as the vocabulary of some document or set of documents. This has many important application in both off-line (indexed) and on-line string matching. More precisely, we have a universe ${\mathbb U}$ of strings, and a non-negative distance function $d: {\mathbb U} \times {\mathbb U} \rightarrow {\mathbb N}$. The distance function is metric, if it satisfies (i) $d(x,y) = 0 ~ \Leftrightarrow ~ x = y$; (ii) d(x,y) = d(y,x); (iii) d(x,y) ≤ d(x,z) + d(z,y). The last item is called the “triangular inequality”, and is the most important property in our case. Many useful distance functions are known to be metric, in particular edit (Levenshtein) distance is metric, which we will use for d.