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
Lectures on Discrete Geometry
An optimal randomized algorithm for maximum Tukey depth
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Approximating Tverberg points in linear time for any fixed dimension
Proceedings of the twenty-eighth annual symposium on Computational geometry
Byzantine vector consensus in complete graphs
Proceedings of the 2013 ACM symposium on Principles of distributed computing
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We present the IteratedTverberg algorithm, the first deterministic algorithm for computing an approximate centerpoint of a set S@?R^d with running time sub-exponential in d. The algorithm is a derandomization of the IteratedRadon algorithm of Clarkson et al. (International Journal of Computational Geometry and Applications 6 (3) (1996) 357-377) and is guaranteed to terminate with an @W(1/d^2)-center. Moreover, it returns a polynomial-time checkable proof of the approximation guarantee, despite the coNP-completeness of testing centerpoints in general. We also explore the use of higher order Tverberg partitions to improve the running time of the deterministic algorithm and improve the approximation guarantee for the randomized algorithm. In particular, we show how to improve the @W(1/d^2)-center of the IteratedRadon algorithm to @W(1/d^r^r^-^1) for a cost of O((rd)^d) in time for any integer r1.