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Dictionary matching with uneven gaps. (English) Zbl 1383.68105
Cicalese, Ferdinando (ed.) et al., Combinatorial pattern matching. 26th annual symposium, CPM 2015, Ischia Island, Italy, June 29 – July 1, 2015. Proceedings. Cham: Springer (ISBN 978-3-319-19928-3/pbk; 978-3-319-19929-0/ebook). Lecture Notes in Computer Science 9133, 247-260 (2015).
Summary: A gap-pattern is a sequence of sub-patterns separated by bounded sequences of don’t care characters (called gaps). A one-gap-pattern is a pattern of the form $$P[\alpha ,\beta ]Q$$, where $$P$$ and $$Q$$ are strings drawn from alphabet $$\varSigma$$ and $$[\alpha , \beta ]$$ are lower and upper bounds on the gap size $$g$$. The gap size $$g$$ is the number of don’t care characters between $$P$$ and $$Q$$. The dictionary matching problem with one-gap is to index a collection of one-gap-patterns, so as to identify all sub-strings of a query text $$T$$ that match with any one-gap-pattern in the collection. Let $${\mathcal D}$$ be such a collection of $$d$$ patterns, where $${\mathcal D}=\{P_i[\alpha _i,\beta _i]Q_i\mid 1\leq i \leq d\}$$. Let $$n=\sum _{i=1}^d|P_i|+|Q_i|$$. Let $$\gamma$$ and $$\lambda$$ be two parameters defined on $${\mathcal D}$$ as follows: $$\gamma = |\{j\mid j \in [\alpha _i,\beta _i], 1\leq i\leq d\}|$$ and $$\lambda = |\{\alpha _i,\beta _i \mid 1\leq i\leq d\}|$$. Specifically $$\gamma$$ is the total number gap lengths possible over all patterns in $${\mathcal D}$$ and $$\lambda$$ is the number of distinct gap boundaries across all the patterns. We present a linear space solution (i.e., $$O(n)$$ words) for answering a dictionary matching query on $${\mathcal D}$$ in time $$O(|T| \gamma \log \lambda \log d+\operatorname{occ})$$, where $$\operatorname{occ}$$ is the output size. The query time can be improved to $$O(|T|\gamma +\operatorname{occ})$$ using $$O(n+d^{1+\epsilon })$$ space, where $$\epsilon >0$$ is an arbitrarily small constant. Additionally, we show a compact/succinct space index offering a space-time trade-off. In the special case where parameters $$\alpha _i$$ and $$\beta _i$$’s for all the patterns are same, our results improve upon the work by A. Amir et al. [Lect. Notes Comput. Sci. 8486, 11–20 (2014; Zbl 1390.68781)]. We also explore several related cases where gaps can occur at arbitrary locations and where gap can be induced in the text rather than pattern.
For the entire collection see [Zbl 1314.68012].

##### MSC:
 68W32 Algorithms on strings 68P05 Data structures
##### Keywords:
dictionary matching; point enclosure queries
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