The input/output complexity of sorting and related problems
Communications of the ACM
Overlaying simply connected planar subdivisions in linear time
Proceedings of the eleventh annual symposium on Computational geometry
An effective way to represent quadtrees
Communications of the ACM
Cache oblivious search trees via binary trees of small height
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Introduction to Algorithms
Exponential Structures for Efficient Cache-Oblivious Algorithms
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
Cache Oblivious Distribution Sweeping
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
Speeding up construction of PMR quadtree-based spatial indexes
The VLDB Journal — The International Journal on Very Large Data Bases
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Box-trees for collision checking in industrial installations
Computational Geometry: Theory and Applications - Special issue on the 18th annual symposium on computational geometry—SoCG2002
I/O-efficient dynamic planar point location
Computational Geometry: Theory and Applications
Foundations of Multidimensional and Metric Data Structures (The Morgan Kaufmann Series in Computer Graphics and Geometric Modeling)
SIAM Journal on Computing
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
External-memory computational geometry
SFCS '93 Proceedings of the 1993 IEEE 34th Annual Foundations of Computer Science
Approximate range searching using binary space partitions
Computational Geometry: Theory and Applications
Approximate range searching in external memory
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Improved bounds on the union complexity of fat objects
FSTTCS '05 Proceedings of the 25th international conference on Foundations of Software Technology and Theoretical Computer Science
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
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We present a new I/O-efficient index structure for storing planar subdivisions. This so-called guard-quadtree is a compressed linear quadtree, which is provably efficient for map overlay, point location, and range queries in low-density subdivisions. In particular, we can preprocess a low-density subdivision with n edges, in O(sort(n)) I/Os, to build a guard-quadtree that allows us to:(i)compute the intersections between the edges of two such preprocessed subdivisions in O(scan(n)) I/Os, where n is the total number of edges in the two subdivisions; (ii)answer a single point location query in O(log"Bn) I/Os, and k batched point location queries in O(scan(n)+sort(k)) I/Os; and (iii)answer range queries for any constant-complexity query range Q in O(1@e(log"Bn)+scan(k"@e)) I/Os for every @e0, where k"@e is the number of edges of the subdivision within distance @e@?diam(Q) from Q. For the special case where the subdivision is a fat triangulation, we show how to obtain the same results with an ordinary (uncompressed) linear quadtree, which we call the star-quadtree. The star-quadtree is fully dynamic and needs only O(log"Bn) I/Os per update. Our algorithms and data structures improve on the previous best known bounds for overlaying general subdivisions, both in the number of I/Os and space usage. The constants in the asymptotic bounds are small, which makes our results applicable in practice. Moreover, our algorithms are simpler than previous approaches and almost all of them are cache-oblivious.