Indexing with gaps

In Indexing with Gaps one seeks to index a text to allow pattern queries that allow gaps within the pattern query. Formally a gapped-pattern over alphabet Σ is a pattern of the form p = p₁g₁p ₂g₂⋯g_ℓp_ℓ+1, where ∀i, p_i ∈ Σ* and each g_i is a gap length ∈ N. Often one considers these patterns with some bound constraints, for example, all gaps are bounded by a gap-bound G. Near-optimal solutions have, lately, been proposed for the case of one gap only with a predetermined size. More specifically, an indexing solution for patterns of the form p ₁·g·p₂, where g is known apriori. In this case the solutions mentioned are preprocessed in O(n log^ε n) time and O(n) space, where the pattern queries are answered in O(|p₁| + |p₂|), for constant sized alphabets. For the more general case when there is a bound G these results can be easily adapted with a multiplicative factor of O(G) for the preprocessing, i.e. O(n log ^ε nG) preprocessing time and O(nG) preprocessing space. Alas, these solutions do not lend to more than one gap. In this paper we propose a solution for k gaps one with preprocessing time O(nG^2k log ^k n log log n) and space of O(nG^2k log^k n) and query time O(m + 2^k log log n), where m = Σ_i=1 |p_i|.