68 research outputs found
On Strong Centerpoints
Let be a set of points in and be a
family of geometric objects. We call a point a strong centerpoint of
w.r.t if is contained in all that
contains more than points from , where is a fixed constant. A
strong centerpoint does not exist even when is the family of
halfspaces in the plane. We prove the existence of strong centerpoints with
exact constants for convex polytopes defined by a fixed set of orientations. We
also prove the existence of strong centerpoints for abstract set systems with
bounded intersection
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.
Small Strong Epsilon Nets
Let P be a set of n points in . A point x is said to be a
centerpoint of P if x is contained in every convex object that contains more
than points of P. We call a point x a strong centerpoint for a
family of objects if is contained in every object that contains more than a constant fraction of points of P. A
strong centerpoint does not exist even for halfspaces in . We
prove that a strong centerpoint exists for axis-parallel boxes in
and give exact bounds. We then extend this to small strong
-nets in the plane and prove upper and lower bounds for
where is the family of axis-parallel
rectangles, halfspaces and disks. Here represents the
smallest real number in such that there exists an
-net of size i with respect to .Comment: 19 pages, 12 figure
Journey to the Center of the Point Set
We revisit an algorithm of Clarkson et al. [K. L. Clarkson et al., 1996], that computes (roughly) a 1/(4d^2)-centerpoint in O~(d^9) time, for a point set in R^d, where O~ hides polylogarithmic terms. We present an improved algorithm that computes (roughly) a 1/d^2-centerpoint with running time O~(d^7). While the improvements are (arguably) mild, it is the first progress on this well known problem in over twenty years. The new algorithm is simpler, and the running time bound follows by a simple random walk argument, which we believe to be of independent interest. We also present several new applications of the improved centerpoint algorithm
Compact finite volume methods for the diffusion equation
An approach to treating initial-boundary value problems by finite volume methods is described, in which the parallel between differential and difference arguments is closely maintained. By using intrinsic geometrical properties of the volume elements, it is possible to describe discrete versions of the div, curl, and grad operators which lead, using summation-by-parts techniques, to familiar energy equations as well as the div curl = 0 and curl grad = 0 identities. For the diffusion equation, these operators describe compact schemes whose convergence is assured by the energy equations and which yield both the potential and the flux vector with second order accuracy. A simplified potential form is especially useful for obtaining numerical results by multigrid and alternating direction implicit (ADI) methods. The treatment of general curvilinear coordinates is shown to result from a specialization of these general results
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