Validating the Knuth-Morris-Pratt Failure Function, Fast and Online
Paweł Gawrychowski, Artur Jeż, Łukasz Jeż · Theory of Computing Systems · 2013
Let $\pi'_{w}$ denote the failure function of the Knuth-Morris-Pratt algorithm for a word w. In this paper we study the following problem: given an integer array $A'[1 \mathinner {\ldotp \ldotp }n]$ , is there a word w over an arbitrary alphabet Σ such that $A'[i]=\pi'_{w}[i]$ for all i? Moreover, what is the minimum cardinality of Σ required? We give an elementary and self-contained $\mathcal{O}(n\log n)$ time algorithm for this problem, thus improving the previously known solution (Duval et al. in Conference in honor of Donald E. Knuth, 2007), which had no polynomial time bound. Using both deeper combinatorial insight into the structure of π′ and advanced algorithmic tools, we further improve the running time to $\mathcal{O}(n)$ .