I/O-efficient map overlay and point location in low-density subdivisions

Authors:
Mark de Berg;Herman Haverkort;Shripad Thite;Laura Toma
Affiliations:
Dept. of Computer Science, Eindhoven University of Technology, The Netherlands;Dept. of Computer Science, Eindhoven University of Technology, The Netherlands;Center for the Mathematics of Information, California Institute of Technology;Dept. of Computer Science, Bowdoin College
Venue:
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Year:
2007

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Abstract

We present improved and simplified 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(sort (n)) I/O's into a compressed linear quadtree such that one can: (i) compute the overlay of two preprocessed subdivisions in O(scan(n)) I/O's, where n is the total number of edges in the two subdivisions, (ii) answer a single point location query in O(logB n) I/O's and k batched point location queries in O(scan(n) + 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(logB 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.