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In this paper, we propose a method for semantic parsing the 3D point cloud of an entire building using a hierarchical approach: first, the raw data is parsed into semantically meaningful spaces (e.g. rooms, etc) that are aligned into a canonical reference coordinate system. Second, the spaces are parsed into their structural and building elements (e.g. walls, columns, etc). Performing these with a strong notation of global 3D space is the backbone of our method. The alignment in the first step injects strong 3D priors from the canonical coordinate system into the second step for discovering elements. This allows diverse challenging scenarios as man-made indoor spaces often show recurrent geometric patterns while the appearance features can change drastically. We also argue that identification of structural elements in indoor spaces is essentially a detection problem, rather than segmentation which is commonly used. We evaluated our method on a new dataset of several buildings with a covered area of over 6, 000m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and over 215 million points, demonstrating robust results readily useful for practical applications.