We compute minimal bases of solutions for a general interpolation problem, a particular case of which is Hermite-Pad\'e approximation. The problem asks to find univariate polynomial relations between $m$ vectors of size $\sigma$; these relations should have small degree with respect to an input degree shift $s \in \mathbb{Z}^m$. For an arbitrary input shift, we propose an algorithm for the computation of an interpolation basis in shifted Popov form with a cost of $O\tilde{~}( m^{w-1} \sigma )$ operations in the base field, where $w$ is the exponent of matrix multiplication, and the notation $O\tilde{~}(\cdot)$ omits logarithmic terms. Earlier works, in the case of Hermite-Pad\'e approximation [Zhou and Labahn, 2012] and in the general interpolation case [Jeannerod et al., 2015], compute non-normalized bases. Since for arbitrary shifts such bases may have size $\Theta( m^2 \sigma )$, the cost bound $O\tilde{~}( m^{w-1} \sigma )$ was feasible only with restrictive assumptions on $s$ for ensuring small output sizes. The question of handling arbitrary shifts with the same complexity bound was left open. To obtain the target cost for any shift, we strengthen the properties of the output bases, and of those obtained during the course of the algorithm: all the bases are computed in shifted Popov form, whose size is always $O( m \sigma )$. Then, we design a divide-and-conquer scheme. We recursively reduce the initial interpolation problem to sub-problems with more convenient shifts by first computing information on the degrees of the intermediary bases.