Optimal, online validation of the pi and pi' failure functions

Let πw denote the failure function of the Morris-Pratt algorithm for a word w. In this paper we study the following problem: given an integer array A[1. . n], is there a word w over arbitrary alphabet Σ such that A[i] = πw[i] for all i? Moreover, what is the minimum cardinal-ity of Σ required? We give a real time linear algorithm for this problem in the unit-cost RAM model with Θ(log n) bits word size. Our algorithm returns a word w over minimal alphabet such that πw = A as well and uses just o(n) words of memory. Then we consider function π instead of π and give an online O(n log n) algorithm for this case. This is the first polynomial algorithm for online version of this problem.