  Algorithmic solutions to the conjugacy problem in the braid groups B_n were
given by Elrifai-Morton in 1994 and by the authors in 1998. Both solutions
yield two conjugacy class invariants which are known as `inf' and `sup'. A
problem which was left unsolved in both papers was the number m of times one
must `cycle' (resp. `decycle') in order to increase inf (resp. decrease sup) or
to be sure that it is already maximal (resp. minimal) for the given conjugacy
class. Our main result is to prove that m is bounded above by n-2 in the
situation of the second algorithm and by ((n^2-n)/2)-1 in the situation of the
first. As a corollary, we show that the computation of inf and sup is
polynomial in both word length and braid index, in both algorithms. The
integers inf and sup determine (but are not determined by) the shortest
geodesic length for elements in a conjugacy class, as defined by Charney, and
so we also obtain a polynomial-time algorithm for computing this geodesic
length.
