An Algorithm to Compute the Character Access Count Distribution for Pattern Matching Algorithms

We propose a framework for the exact probabilistic analysis of window-based pattern matching algorithms, such as Boyer--Moore, Horspool, Backward DAWG Matching, Backward Oracle Matching, and more. In particular, we develop an algorithm that efficiently computes the distribution of a pattern matching algorithm's running time cost (such as the number of text character accesses) for any given pattern in a random text model. Text models range from simple uniform models to higher-order Markov models or hidden Markov models (HMMs). Furthermore, we provide an algorithm to compute the exact distribution of \emph{differences} in running time cost of two pattern matching algorithms. Methodologically, we use extensions of finite automata which we call \emph{deterministic arithmetic automata} (DAAs) and \emph{probabilistic arithmetic automata} (PAAs)~\cite{Marschall2008}. Given an algorithm, a pattern, and a text model, a PAA is constructed from which the sought distributions can be derived using dynamic programming. To our knowledge, this is the first time that substring- or suffix-based pattern matching algorithms are analyzed exactly by computing the whole distribution of running time cost. Experimentally, we compare Horspool's algorithm, Backward DAWG Matching, and Backward Oracle Matching on prototypical patterns of short length and provide statistics on the size of minimal DAAs for these computations.