Computational geometry: an introduction
Computational geometry: an introduction
SCG '85 Proceedings of the first annual symposium on Computational geometry
An efficient algorithm for computing the maximum empty rectangle in three dimensions
Information Sciences—Applications: An International Journal
Efficient computation of temporal aggregates with range predicates
PODS '01 Proceedings of the twentieth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Efficient aggregation over objects with extent
Proceedings of the twenty-first ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
CRB-Tree: An Efficient Indexing Scheme for Range-Aggregate Queries
ICDT '03 Proceedings of the 9th International Conference on Database Theory
Efficient Execution of Range-Aggregate Queries in Data Warehouse Environments
ER '01 Proceedings of the 20th International Conference on Conceptual Modeling: Conceptual Modeling
Scaling and related techniques for geometry problems
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
Approximating extent measures of points
Journal of the ACM (JACM)
Range Aggregate Processing in Spatial Databases
IEEE Transactions on Knowledge and Data Engineering
Optimizing spatial Min/Max aggregations
The VLDB Journal — The International Journal on Very Large Data Bases
Range mode and range median queries on lists and trees
Nordic Journal of Computing
Algorithms for range-aggregate query problems involving geometric aggregation operations
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Approximate range mode and range median queries
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
On the power of the semi-separated pair decomposition
Computational Geometry: Theory and Applications
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A fundamental and well-studied problem in computational geometry is range searching, where the goal is to preprocess a set, S, of geometric objects (e.g., points in the plane) so that the subset S^'@?S that is contained in a query range (e.g., an axes-parallel rectangle) can be reported efficiently. However, in many situations, what is of interest is to generate a more informative ''summary'' of the output, obtained by applying a suitable aggregation function on S^'. Examples of such aggregation functions include count, sum, min, max, mean, median, mode, and top-k that are usually computed on a set of weights defined suitably on the objects. Such range-aggregate query problems have been the subject of much recent research in both the database and the computational geometry communities. In this paper, we further generalize this line of work by considering aggregation functions on point-sets that measure the extent or ''spread'' of the objects in the retrieved set S^'. The functions considered here include closest pair, diameter, and width. The challenge here is that these aggregation functions (unlike, say, count) are not efficiently decomposable in the sense that the answer to S^' cannot be inferred easily from answers to subsets that induce a partition of S^'. Nevertheless, we have been able to obtain space- and query-time-efficient solutions to several such problems including: closest pair queries with axes-parallel rectangles on point sets in the plane and on random point-sets in R^d (d=2), closest pair queries with disks on random point-sets in the plane, diameter queries on point-sets in the plane, and guaranteed-quality approximations for diameter and width queries in the plane. Our results are based on a combination of geometric techniques, including multilevel range trees, Voronoi Diagrams, Euclidean Minimum Spanning Trees, sparse representations of candidate outputs, and proofs of (expected) upper bounds on the sizes of such representations.