On k-hulls and related problems
SIAM Journal on Computing
Approximations and optimal geometric divide-and-conquer
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Discrete Mathematics - Topological, algebraical and combinatorial structures; Froli´k's memorial volume
Points, spheres, and separators: a unified geometric approach to graph partitioning
Points, spheres, and separators: a unified geometric approach to graph partitioning
Approximate center points in dense point sets
Information Processing Letters
Geometric Mesh Partitioning: Implementation and Experiments
SIAM Journal on Scientific Computing
An optimal randomized algorithm for maximum Tukey depth
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Centerpoints and Tverberg's technique
Computational Geometry: Theory and Applications
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We present the Iterated-Tverberg algorithm, the first deterministic algorithm for computing an approximate centerpoint of a set S ∈ Rd with running time sub-exponential in d. The algorithm is a derandomization of the Iterated-Radon algorithm of Clarkson et al and is guaranteed to terminate with an O(1/d2)-center. Moreover, it returns a polynomial-time checkable proof of the approximation guarantee, despite the coNP-Completenes of testing centerpoints in general. We also explore the use of higher order Tverberg partitions to improve the runtime of the deterministic algorithm and improve the approximation guarantee for the randomized algorithm. In particular, we show how to improve the O(1/d2)-center of the Iterated-Radon algorithm to O(1/dr/(r-1)) for a cost of O((rd)d) in time for any integer r.