New upper bounds in Klee's measure problem
SIAM Journal on Computing
Concept learning with geometric hypotheses
COLT '95 Proceedings of the eighth annual conference on Computational learning theory
Journal of Computer and System Sciences
Computing the discrepancy with applications to supersampling patterns
ACM Transactions on Graphics (TOG)
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
Handbook of discrete and computational geometry
Efficient and small representation of line arrangements with applications
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
The discrepancy method: randomness and complexity
The discrepancy method: randomness and complexity
Algorithmics for Hard Problems
Algorithmics for Hard Problems
The Precision of Query Points as a Resource for Learning Convex Polytopes with Membership Queries
COLT '00 Proceedings of the Thirteenth Annual Conference on Computational Learning Theory
Translating a regular grid over a point set
Computational Geometry: Theory and Applications - Special issue: The European workshop on computational geometry -- CG01
An algorithm for point cluster generalization based on the Voronoi diagram
Computers & Geosciences
Mesh simplification for building typification
International Journal of Geographical Information Science
An information-based exploration strategy for environment mapping with mobile robots
Robotics and Autonomous Systems
Geo-Enabling spatially relevant data for mobile information use and visualisation
W2GIS'05 Proceedings of the 5th international conference on Web and Wireless Geographical Information Systems
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Dot maps--drawings of point sets--are a well known cartographic method to visualize density functions over an area. We study the problem of simplifying a given dot map: given a set P of points in the plane, we want to compute a smaller set Q of points whose distribution approximates the distribution of the original set P.We formalize this using the concept of ε-approximations, and we give efficient algorithms for computing the approximation error of a set Q of m points with respect to a set P of n points (with m ≤ n) for certain families of ranges, namely unit squares, arbitrary squares, and arbitrary rectangles.If the family R of ranges is the family of all possible unit squares, then we compute the approximation error of Q with respect to P in O(n log n) time. If R is the family of all possible rectangles, we present an O(mn log n) time algorithm. If R is the family of all possible squares, then we present a simple O(m2n + n log n) algorithm and an O(n2 √n log n) time algorithm which is more efficient in the worst case.Finally, we develop heuristics to compute good approximations, and we evaluate our heuristics experimentally.