Fast String Dictionary Lookup with One Error

Abstract

A set of strings, called a string dictionary, is a basic string data structure. The most primitive query, where one seeks the existence of a pattern in the dictionary, is called a lookup query. Approximate lookup queries, i.e., to lookup the existence of a pattern with a bounded number of errors, is a fundamental string problem. Several data structures have been proposed to do so efficiently. Almost all solutions consider a single error, as will this result. Lately, Belazzougui and Venturini (CPM 2013) raised the question whether one can construct efficient indexes that support lookup queries with one error in optimal query time, that is, $O(|p|/\omega + \textit{occ})$, where $p$ is the query, $\omega$ the machine word-size, and $occ$ the number of occurrences.

Specifically, for the problem of one mismatch and constant alphabet size, we obtain optimal query time. For a dictionary of $d$ strings our proposed index uses $O(\omega d\log ^{1+\epsilon }d)$ additional bit space (beyond the space required to access the dictionary data, which can be maintained in compressed form). Our results are parameterized for a space-time tradeoff.

We propose more results for the case of lookup queries with one insertion/deletion on dictionaries over a constant sized alphabet. These results are especially effective for large patterns.

T. Chan—The research is supported by an NSERC grant.

M. Lewenstein—This research is supported by a BSF grant 2010437 and a GIF grant 1147/2011.