210 research outputs found
Approximating Tverberg Points in Linear Time for Any Fixed Dimension
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).Comment: 14 pages, 2 figures. A preliminary version appeared in SoCG 2012.
This version removes an incorrect example at the end of Section 3.
Regression Depth and Center Points
We show that, for any set of n points in d dimensions, there exists a
hyperplane with regression depth at least ceiling(n/(d+1)). as had been
conjectured by Rousseeuw and Hubert. Dually, for any arrangement of n
hyperplanes in d dimensions there exists a point that cannot escape to infinity
without crossing at least ceiling(n/(d+1)) hyperplanes. We also apply our
approach to related questions on the existence of partitions of the data into
subsets such that a common plane has nonzero regression depth in each subset,
and to the computational complexity of regression depth problems.Comment: 14 pages, 3 figure
Byzantine Vector Consensus in Complete Graphs
Consider a network of n processes each of which has a d-dimensional vector of
reals as its input. Each process can communicate directly with all the
processes in the system; thus the communication network is a complete graph.
All the communication channels are reliable and FIFO (first-in-first-out). The
problem of Byzantine vector consensus (BVC) requires agreement on a
d-dimensional vector that is in the convex hull of the d-dimensional input
vectors at the non-faulty processes. We obtain the following results for
Byzantine vector consensus in complete graphs while tolerating up to f
Byzantine failures:
* We prove that in a synchronous system, n >= max(3f+1, (d+1)f+1) is
necessary and sufficient for achieving Byzantine vector consensus.
* In an asynchronous system, it is known that exact consensus is impossible
in presence of faulty processes. For an asynchronous system, we prove that n >=
(d+2)f+1 is necessary and sufficient to achieve approximate Byzantine vector
consensus.
Our sufficiency proofs are constructive. We show sufficiency by providing
explicit algorithms that solve exact BVC in synchronous systems, and
approximate BVC in asynchronous systems.
We also obtain tight bounds on the number of processes for achieving BVC
using algorithms that are restricted to a simpler communication pattern
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