We study log Calabi--Yau varieties obtained as a blow-up of a toric variety along hypersurfaces in its toric boundary. Mirrors to such varieties are constructed by Gross and Siebert from a canonical scattering diagram built by using punctured Gromov--Witten invariants of Abramovich, Chen, Gross and Siebert. We show that there is a piecewise-linear isomorphism between the canonical scattering diagram and a scattering diagram defined algorithmically, following a higher-dimensional generalization of the Kontsevich--Soibelman construction. We deduce that the punctured Gromov--Witten invariants of the log Calabi--Yau variety can be captured from this algorithmic construction. This generalizes previous results of Gross, Pandharipande and Siebert on ``the tropical vertex'' to higher dimensions. As a particular example, we compute these invariants for a nontoric blow-up of the three-dimensional projective space along two lines.
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