String indexing for patterns with wildcards. Theory Comput

We consider the problem of indexing a string t of length n to report the occurrences of a query pattern p containing m characters and j wildcards. Let occ be the number of occurrences of p in t, and σ the size of the alphabet. We obtain the following results. • A linear space index with query time O(m+σj log log n+occ). This significantly improves the previously best known linear space index by Lam et al. [ISAAC 2007], which requires query time Θ(jn) in the worst case. • An index with query time O(m + j + occ) using space O(σk2n logk log n), where k is the maximum number of wildcards allowed in the pattern. This is the first non-trivial bound with this query time. • A time-space trade-off, generalizing the index by Cole et al. [STOC 2004]. We also show that these indexes can be generalized to allow variable length gaps in the pattern. Our results are obtained using a novel combination of well-known and new techniques, which could be of independent interest.