We present a quadtree-based decomposition of the interior of a polygon with holes. The complete decomposition yields a constant factor approximation of the minimum weight Steiner triangulation (MWST) of the polygon. We show that this approximate MWST supports ray shooting queries in the query-sensitive sense as defined by Mitchell, Mount and Suri. A proper truncation of our quadtree-based decomposition yields another constant factor approximation of the MWST. For a polygon with n vertices, the complexity of this approximate MWST is O(n log n) and it can be constructed in O(n log n) time. The running time is optimal in the algebraic decision tree model.
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