Approximating Tverberg Points in Linear Time for Any Fixed Dimension
Abstract
Let P be a d-dimensional n-point set. A Tverberg-partition of P is a partition of P into r sets P_1, ..., P_r such that the convex hulls conv(P_1), ..., conv(P_r) have non-empty intersection. A point in the intersection of the conv(P_i)'s is called a Tverberg point of depth r for P. A classic result by Tverberg implies that there always exists a Tverberg partition of size n/(d+1), but it is not known how to find such a partition in polynomial time. Therefore, approximate solutions are of interest. We describe a deterministic algorithm that finds a Tverberg partition of size n/4(d+1)^3 in time d^{O(log d)} n. This means that for every fixed dimension we can compute an approximate Tverberg point (and hence also an approximate centerpoint) in linear time. Our algorithm is obtained by combining a novel lifting approach with a recent result by Miller and Sheehy (2010).
Cite
@article{arxiv.1107.0104,
title = {Approximating Tverberg Points in Linear Time for Any Fixed Dimension},
author = {Wolfgang Mulzer and Daniel Werner},
journal= {arXiv preprint arXiv:1107.0104},
year = {2020}
}
Comments
14 pages, 2 figures. A preliminary version appeared in SoCG 2012. This version removes an incorrect example at the end of Section 3.1