Quadtree, ray shooting and approximate minimum weight Steiner triangulation

Authors:
Siu-Wing Cheng;Kam-Hing Lee
Affiliations:
Department of Computer Science, HKUST, Clear Water Bay, Hong Kong, China;eCyberPay.com Limited, 25/F, CEF Lend Lease Plaza, 663 King's Road, Hong Kong, China
Venue:
Computational Geometry: Theory and Applications
Year:
2002

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Abstract

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.