  The purpose of the present paper is to study the computation complexity of
deep ReLU neural networks to approximate functions in H\"older-Nikol'skii
spaces of mixed smoothness $H_\infty^\alpha(\mathbb{I}^d)$ on the unit cube
$\mathbb{I}^d:=[0,1]^d$. In this context, for any function $f\in
H_\infty^\alpha(\mathbb{I}^d)$, we explicitly construct nonadaptive and
adaptive deep ReLU neural networks having an output that approximates $f$ with
a prescribed accuracy $\varepsilon$, and prove dimension-dependent bounds for
the computation complexity of this approximation, characterized by the size and
the depth of this deep ReLU neural network, explicitly in $d$ and
$\varepsilon$. Our results show the advantage of the adaptive method of
approximation by deep ReLU neural networks over nonadaptive one.
