Pattern Matching in Constrained Sequences

Constrained sequences find applications in communication, magnetic recording, and biology. In this paper, we restrict our attention to the so-called (d, k) constrained binary sequences in which any run of zeros must be of length at least d and at most k, where 0lesd<k. In some applications one needs to know the number of occurrences of a given pattern w in such sequences, for which we coin the term constrained pattern matching. For a given word w or a set of words W, we estimate the (conditional) probability of the number of occurrences of w in a (d, k) sequence generated by a memoryless source. As a by-product, we enumerate asymptotically the number of (d, k) sequences with exactly r occurrences of a given word w, and compute Shannon entropy of (d, k) sequences with a given number of occurrences of w. Throughout this paper we use techniques of analytic information theory such as combinatorial calculus, generating functions, and complex asymptotics.