String Indexing for Patterns with Wildcards

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 $\sigma$ the size of the alphabet. We obtain the following results. - A linear space index with query time $O(m+\sigma^j \log \log n + occ)$. This significantly improves the previously best known linear space index by Lam et al. [ISAAC 2007], which requires query time $\Theta(jn)$ in the worst case. - An index with query time $O(m+j+occ)$ using space $O(\sigma^{k^2} n \log^k \log n)$, 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

A.: Dotted suffix trees: a structure for approximate text indexing

Abstract. In this work, we address is text indexing for approximate matching. Given a text T which undergoes some preprocessing to generate an index, we can later query this index to identify the places where a string occurs up to a certain number of errors k (edition distance). The indexing structure occupies space O(n log k n) in the average case, independent of alphabet size. This structure can be used to report the existence of a match with k errors in O(3 k m k+1) and to report the occurrences in O(3 k m k+1 + ed) time, where m is the length of the pattern and where ed the number of matching edit scripts. The construction of the structure has time bound by O(kN|Σ|), where N is the number of nodes in the index and |Σ | the alphabet size