  The algorithm given by Akbulut-Ozbagci constructs an explicit open book
decomposition on a contact three-manifold described by a contact surgery on a
link in the three-sphere. In this article, we will improve this algorithm by
using Giroux's contact cell decomposition process. Our algorithm is more
economical on choosing the supporting genus of the open book; in particular it
gives a good upper bound for the recently defined ``minimal supporting genus
invariant'' of contact structures.
