Complexity of the Havas, Majewski, Matthews LLL Hermite Normal Form Algorithm

We consider the complexity of the LLL HNF algorithm Havaset al.(1998, Algorithm 4). This algorithm takes as input an m by n matrix G of integers and produces as output a matrixb GLm(Z) so thatA = bG is in Hermite normal form (upside down). The analysis is similar to that of an extended LLL algorithm as given in van der Kallen (1998).