Before revision:  for any function f\in H^\alpha_\infty( \mathbb I ^d), we explicitly construct a deep ReLU neural network having an output that approximates f in the W_0^{1,p \mathbb I ^d)-norm with a prescribed accuracy \varepsilon, and prove tight dimension-dependent  bounds of the computation complexity of this approximation, characterized as the number of weights and the depth of this deep ReLU neural network, explicitly in d and \varepsilon.
After revision:  for any function f\in H^\alpha_\infty( \mathbb I of mixed smoothness, we explicitly construct a deep ReLU neural network having an output that approximates f in the W_0^{1,p \mathbb I ^d)-norm with a prescribed accuracy \varepsilon, and prove tight dimension-dependent bounds of the computation complexity of this approximation, characterized as the number of weights and the depth of this deep ReLU neural network, explicitly in d and \varepsilon.