  We show that the absolute worst case time complexity for Hopcroft's
minimization algorithm applied to unary languages is reached only for de Bruijn
words. A previous paper by Berstel and Carton gave the example of de Bruijn
words as a language that requires O(n log n) steps by carefully choosing the
splitting sets and processing these sets in a FIFO mode. We refine the previous
result by showing that the Berstel/Carton example is actually the absolute
worst case time complexity in the case of unary languages. We also show that a
LIFO implementation will not achieve the same worst time complexity for the
case of unary languages. Lastly, we show that the same result is valid also for
the cover automata and a modification of the Hopcroft's algorithm, modification
used in minimization of cover automata.
