Abstract
We present improved and simplified \({\textsc{i/o}}\)-efficient algorithms for two problems on planar low-density subdivisions, namely map overlay and point location. More precisely, we show how to preprocess a low-density subdivision with n edges in \(O({\mathit{sort}}(n))\) i/o’s into a compressed linear quadtree such that one can:
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compute the overlay of two preprocessed subdivisions in \(O({\mathit{scan}}(n))\) i/o’s, where n is the total number of edges in the two subdivisions,
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answer a single point location query in O(log B n) i/o’s and k batched point location queries in \(O({\mathit{scan}}(n) + {\mathit{sort}}(k))\) i/o’s.
For the special case where the subdivision is a fat triangulation, we show how to obtain the same bounds with an ordinary (uncompressed) quadtree, and we show how to make the structure fully dynamic using O(log B n) i/o’s per update. Our algorithms and data structures improve on the previous best known bounds for general subdivisions both in the number of i/o’s and storage usage, they are significantly simpler, and several of our algorithms are cache-oblivious.
MdB and ST were supported by the Netherlands’ Organisation for Scientific Research (NWO).
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de Berg, M., Haverkort, H., Thite, S., Toma, L. (2007). I/O-Efficient Map Overlay and Point Location in Low-Density Subdivisions. In: Tokuyama, T. (eds) Algorithms and Computation. ISAAC 2007. Lecture Notes in Computer Science, vol 4835. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77120-3_44
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DOI: https://doi.org/10.1007/978-3-540-77120-3_44
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