Continuant polynomials and worst-case behavior of Hopcroft's minimization algorithm

This paper is concernedwith the analysis of theworst case behavior of Hopcroft’s algorithm for minimizing deterministic finite state automata. We extend a result of Castiglione, Restivo and Sciortino. They show that Hopcroft’s algorithm has a worst case behavior for the automata recognizing Fibonacci words. In a previous paper, we have proved that this holds for all standard Sturmian words having an ultimately periodic directive sequence (the directive sequence for Fibonacci words is (1, 1, . . .)). We prove here that the same conclusion holds for all standard Sturmian words having a directive sequence with bounded elements. More precisely, we obtain in fact a characterization of those directive sequences for which Hopcroft’s algorithm hasworst case running time. These are the directive sequences (d1, d2, d3, . . .) for which the sequence of geometric means (d1d2 · · · dn) is bounded. As a consequence, we easily show that there exist directive sequences for which the worst case for the running time is not attained. © 2009 Elsevier B.V. All rights reserved.