The influence of epitaxial strain on the surface and inter-layer diffusions are investigated using molecular statics and transition state theory with several types of embedded atom method potentials for Ag and Ni. Quantitatively to analyze the competition of the surface and inter-layer diffusions in the instability to the multi-layer growth, Ehrlich-Schwoebel barrier and the attempt frequencies of the surface and inter-layer diffusions by both hopping and exchange mechanisms are considered simultaneously. The attempt frequencies of exchange mechanism are larger by order one than those of hopping mechanism and especially, the difference becomes more severe for the inter-layer diffusion. Considering both the attempt frequency and activation energy barrier shows that the layer-by-layer growth is enhanced by compressive strain for Ag(0 0 1), which is confirmed by the existing linear stability theory and kinetic Monte Carlo simulations. (c) 2006 Elsevier B.V. All rights reserved.