Computational geometry: an introduction
Computational geometry: an introduction
Skip lists: a probabilistic alternative to balanced trees
Communications of the ACM
Introduction to algorithms
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
SIGMOD '99 Proceedings of the 1999 ACM SIGMOD international conference on Management of data
The space complexity of approximating the frequency moments
Journal of Computer and System Sciences
The C++ standard library: a tutorial and reference
The C++ standard library: a tutorial and reference
Space-efficient online computation of quantile summaries
SIGMOD '01 Proceedings of the 2001 ACM SIGMOD international conference on Management of data
Models and issues in data stream systems
Proceedings of the twenty-first ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
An Approximate L1-Difference Algorithm for Massive Data Streams
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Approximating extent measures of points
Journal of the ACM (JACM)
Coresets in dynamic geometric data streams
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Sampling algorithms in a stream operator
Proceedings of the 2005 ACM SIGMOD international conference on Management of data
Faster core-set constructions and data-stream algorithms in fixed dimensions
Computational Geometry: Theory and Applications
Data streams: algorithms and applications
Foundations and Trends® in Theoretical Computer Science
A space-optimal data-stream algorithm for coresets in the plane
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Approximate order-k Voronoi cells over positional streams
Proceedings of the 15th annual ACM international symposium on Advances in geographic information systems
Summarizing spatial data streams using ClusterHulls
Journal of Experimental Algorithmics (JEA)
Simplified Planar Coresets for Data Streams
SWAT '08 Proceedings of the 11th Scandinavian workshop on Algorithm Theory
Approximate voronoi cell computation on spatial data streams
The VLDB Journal — The International Journal on Very Large Data Bases
Answering linear optimization queries with an approximate stream index
Knowledge and Information Systems
Faster core-set constructions and data-stream algorithms in fixed dimensions
Computational Geometry: Theory and Applications
Approximate ellipsoid in the streaming model
COCOA'10 Proceedings of the 4th international conference on Combinatorial optimization and applications - Volume Part II
Adaptive spatial partitioning for multidimensional data streams
ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
Streaming algorithms for geometric problems
FSTTCS'04 Proceedings of the 24th international conference on Foundations of Software Technology and Theoretical Computer Science
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Geometric coordinates are an integral part of many data streams. Examples include sensor locations in environmental monitoring, vehicle locations in traffic monitoring or battlefield simulations, scientific measurements of earth or atmospheric phenomena, etc. How can one summarize such data streams using limited storage so that many natural geometric queries can be answered faithfully? Some examples of such queries are: report the smallest convex region in which a chemical leak has been sensed, or track the diameter of the dataset. One can also pose queries over multiple streams: track the minimum distance between the convex hulls of two data streams; or report when datasets A and B are no longer linearly separable.In this paper, we propose an adaptive sampling scheme that gives provably optimal error bounds for extremal problems of this nature. All our results follow from a single technique for computing the approximate convex hull of a point stream in a single pass. Our main result is this: given a stream of two-dimensional points and an integer r, we can maintain an adaptive sample of at most 2r + 1 points such that the distance between the true convex hull and the convex hull of the sample points is O(D/r2), where D is the diameter of the sample set. With our sample convex hull, all the queries mentioned above can be answered in either O(log r) or O(r) time.