Multiple serial episode matching

In [BCGM01] we have generalized the Knuth-Morris-Pratt (KMP) pattern matching algorithm and defined a nonconventional kind of RAM, the MP–RAMs which model more closely the microprocessor operations, and designed an O(n) on-line algorithm for solving the serial episode matching problem on MP–RAMs when there is only one single episode. We here give two extensions of this algorithm to the case when we search for several patterns simultaneously and compare them. More preciseley, given q + 1 strings (a text t of length n and q patterns m1, . . . , mq) and a natural number w, the multiple serial episode matching problem consists in finding the number of size w windows of text t which contain patterns m1, . . . , mq as subsequences, i.e. for each mi, if mi = p1, . . . , pk, the letters p1, . . . , pk occur in the window, in the same order as in mi, but not necessarily consecutively (they may be interleaved with other letters).