468 research outputs found
Halving Balls in Deterministic Linear Time
Let \D be a set of pairwise disjoint unit balls in and the
set of their center points. A hyperplane \Hy is an \emph{-separator} for
\D if each closed halfspace bounded by \Hy contains at least points
from . This generalizes the notion of halving hyperplanes, which correspond
to -separators. The analogous notion for point sets has been well studied.
Separators have various applications, for instance, in divide-and-conquer
schemes. In such a scheme any ball that is intersected by the separating
hyperplane may still interact with both sides of the partition. Therefore it is
desirable that the separating hyperplane intersects a small number of balls
only. We present three deterministic algorithms to bisect or approximately
bisect a given set of disjoint unit balls by a hyperplane: Firstly, we present
a simple linear-time algorithm to construct an -separator for balls
in , for any , that intersects at most
balls, for some constant that depends on and . The number of
intersected balls is best possible up to the constant . Secondly, we present
a near-linear time algorithm to construct an -separator in
that intersects balls. Finally, we give a linear-time algorithm to
construct a halving line in that intersects
disks.
Our results improve the runtime of a disk sliding algorithm by Bereg,
Dumitrescu and Pach. In addition, our results improve and derandomize an
algorithm to construct a space decomposition used by L{\"o}ffler and Mulzer to
construct an onion (convex layer) decomposition for imprecise points (any point
resides at an unknown location within a given disk)
Memoryless Worker-Task Assignment with Polylogarithmic Switching Cost
We study the basic problem of assigning memoryless workers to tasks with
dynamically changing demands. Given a set of workers and a multiset of tasks, a memoryless worker-task assignment function is
any function that assigns the workers to the tasks based only
on the current value of . The assignment function is said to have
switching cost at most if, for every task multiset , changing the
contents of by one task changes by at most worker
assignments. The goal of memoryless worker task assignment is to construct an
assignment function with the smallest possible switching cost.
In past work, the problem of determining the optimal switching cost has been
posed as an open question. There are no known sub-linear upper bounds, and
after considerable effort, the best known lower bound remains 4 (ICALP 2020).
We show that it is possible to achieve polylogarithmic switching cost. We
give a construction via the probabilistic method that achieves switching cost
and an explicit construction that achieves switching cost
. We also prove a super-constant lower bound on
switching cost: we show that for any value of , there exists a value of
for which the optimal switching cost is . Thus it is not possible to achieve
a switching cost that is sublinear strictly as a function of .
Finally, we present an application of the worker-task assignment problem to a
metric embeddings problem. In particular, we use our results to give the first
low-distortion embedding from sparse binary vectors into low-dimensional
Hamming space.Comment: ICALP 202
Unions of Onions: Preprocessing Imprecise Points for Fast Onion Decomposition
Let be a set of pairwise disjoint unit disks in the plane.
We describe how to build a data structure for so that for any
point set containing exactly one point from each disk, we can quickly find
the onion decomposition (convex layers) of .
Our data structure can be built in time and has linear size.
Given , we can find its onion decomposition in time, where
is the number of layers. We also provide a matching lower bound. Our solution
is based on a recursive space decomposition, combined with a fast algorithm to
compute the union of two disjoint onionComment: 10 pages, 5 figures; a preliminary version appeared at WADS 201
Subsampling in Smoothed Range Spaces
We consider smoothed versions of geometric range spaces, so an element of the
ground set (e.g. a point) can be contained in a range with a non-binary value
in . Similar notions have been considered for kernels; we extend them to
more general types of ranges. We then consider approximations of these range
spaces through -nets and -samples (aka
-approximations). We characterize when size bounds for
-samples on kernels can be extended to these more general
smoothed range spaces. We also describe new generalizations for -nets to these range spaces and show when results from binary range spaces can
carry over to these smoothed ones.Comment: This is the full version of the paper which appeared in ALT 2015. 16
pages, 3 figures. In Algorithmic Learning Theory, pp. 224-238. Springer
International Publishing, 201
Lower bounds for k-distance approximation
Consider a set P of N random points on the unit sphere of dimension ,
and the symmetrized set S = P union (-P). The halving polyhedron of S is
defined as the convex hull of the set of centroids of N distinct points in S.
We prove that after appropriate rescaling this halving polyhedron is Hausdorff
close to the unit ball with high probability, as soon as the number of points
grows like . From this result, we deduce probabilistic lower
bounds on the complexity of approximations of the distance to the empirical
measure on the point set by distance-like functions
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