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The reconstruction of an Ag monolayer on Ag/Pt(111) is analyzed theoretically, employing a vertically extended Frenkel-Kontorova model whose parameters are derived from density-functional theory. Energy minimization is carried out using simulated quantum annealing techniques. Our results are compatible with the scanning tunneling microscopy experiments, where a striped pattern is initially found which transforms into a triangular reconstruction upon annealing. In our model we recognize the first structure as a metastable state, while the second one is the true energy minimum.