Erratum to the Paper “a Note on the Factorization Theorem of Toric Birational Maps after Morelli and Its Toroidal Extension”

1. the failure of the algorithm in [AMR] and [Morelli1] for the strong factorization pointed out by Kalle Karu, 2. the statement of a refined weak factorization theorem for toroidal birational morphisms in [AMR], in the form utilized in [AKMR] for the proof of the weak factorization theorem for general birationla maps, avoiding the use of the above mentioned algorithm for the strong factorization, and 3. a list of corrections for a few other mistakes in [AMR], mostly pointed out by Laurent Bonavero. We would like to emphasize that though [AMR] is a joint work with D. Abramovich and S. Rashid, the author of this erratum, Kenji Matsuki, is solely responsible for all the errors above.