Efficient index for weighted sequences

The problem of finding factors of a text string which are identical or similar to a given pattern string is a central problem in computer science. A generalised version of this problem consists in implementing an index over the text to support efficient on-line pattern queries. We study this problem in the case where the text is weighted: for every position of the text and every letter of the alphabet a probability of occurrence of this letter at this position is given. Sequences of this type, also called position weight matrices, are commonly used to represent imprecise or uncertain data. A weighted sequence may represent many different strings, each with probability of occurrence equal to the product of probabilities of its letters at subsequent positions. Given a probability threshold 1/z, we say that a pattern string P matches a weighted text at starting position i if the product of probabilities of the letters of P at positions i, . . . , i + |P| - 1 in the text is at least 1/z. In this article, we present an O(nz)-time construction of an O(nz)-sized index that can answer pattern matching queries in a weighted text over a constant-sized alphabet in optimal time. This improves upon the state of the art by a factor of z log z in construction time and space. Other applications of this data structure include an O(nz)-time construction of the weighted prefix table and an O(nz)-time computation of all covers of a weighted sequence, which improve upon the time complexity of the state of the art by the same factor.