Internal Pattern Matching Queries in a Text and Applications

We consider several types of internal queries: questions about subwords of a text. As the main tool we develop an optimal data structure for the problem called here internal pattern matching. This data structure provides constant-time answers to queries about occurrences of one subword x in another subword y of a given text, assuming that |y| = O(|x|), which allows for a constant-space representation of all occurrences. This problem can be viewed as a natural extension of the well-studied pattern matching problem. The data structure has linear size and admits a linear-time construction algorithm. Using the solution to the internal pattern matching problem, we obtain very efficient data structures answering queries about: primitivity of subwords, periods of subwords, general substring compression, and cyclic equivalence of two subwords. All these results improve upon the best previously known counterparts. The linear construction time of our data structure also allows to improve the algorithm for finding δ-subrepetitions in a text (a more general version of maximal repetitions, also called runs). For any fixed δ we obtain the first linear-time algorithm, which matches the linear time complexity of the algorithm computing runs. Our data structure has already been used as a part of the efficient solutions for subword suffix rank & selection, as well as substring compression using Burrows-Wheeler transform composed with run-length encoding. The model of internal queries in texts is connected to the well-studied problem of text indexing. Both models have their origins in the introduction of suffix trees. However, there is an important difference: in our model the size of the representation of a query is constant and therefore enables faster query time. Our results can be viewed as efficient solutions to “internal” equivalents of several basic problems of regular pattern matching and make an improvement in a majority of already published results related to internal queries.