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Abstract

Let π'_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 .. n], is there a word w over an arbitrary alphabet Σ such that A'[i]=π'_w[i] for all i? Moreover, what is the minimum cardinality of Σ required? We give an elementary and self-contained \mathcalO}(nłog n) time algorithm for this problem, thus improving the previously known solution, 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).