Quasi-optimal range searching in spaces of finite VC-dimension
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Quasi-optimal upper bounds for simplex range searching and new zone theorems
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Discrete & Computational Geometry - Special issue on ACM symposium on computational geometry, North Conway
Cutting hyperplanes for divide-and-conquer
Discrete & Computational Geometry
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
An on-line algorithm for fitting straight lines between data ranges
Communications of the ACM
Exploiting duality in summarization with deterministic guarantees
Proceedings of the 13th ACM SIGKDD international conference on Knowledge discovery and data mining
Approximating Points by a Piecewise Linear Function: I
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Approximating Points by a Piecewise Linear Function: II. Dealing with Outliers
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
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Given a real ε 0, an integer g ≥ 0 and a set of points in the plane, we study the problem of computing a piecewise linear functional curve with minimum number of line segments to approximate all points after removing g outliers such that the approximation error is at most ε. We give an improved algorithm over the previous work. The algorithm is based on two dynamic data structures developed in this paper for the simplicial thickness queries, which are of independent interest. For a set S of simplices in the d-D space Ed (d ≥ 2 is a constant), the simplicial thickness of a point p is defined as the number of simplices in S that contain p. Given a set P of n points in Ed, we develop two linear-space dynamic data structures to support the following operations. (1) Simplex insertion: Insert a simplex into S. (2) Simplex deletion: Delete a simplex from S. (3) Simplicial thickness query: Given a query simplex σ, compute the minimum simplicial thickness among all points in σ ∩ P. The first data structure supports each operation in O(n1-1/d) time with O(n1+δ) time preprocessing, for any constant δ 0; the second one supports each operation in O(n1-1/d(log n)O(1)) time with O(n log n) time preprocessing. These data structures may also find other applications.