468 research outputs found
On the Complexity of the k-Level in Arrangements of Pseudoplanes
A classical open problem in combinatorial geometry is to obtain tight
asymptotic bounds on the maximum number of k-level vertices in an arrangement
of n hyperplanes in d dimensions (vertices with exactly k of the hyperplanes
passing below them). This is a dual version of the k-set problem, which, in a
primal setting, seeks bounds for the maximum number of k-sets determined by n
points in d dimensions, where a k-set is a subset of size k that can be
separated from its complement by a hyperplane. The k-set problem is still wide
open even in the plane, with a substantial gap between the best known upper and
lower bounds. The gap gets larger as the dimension grows. In three dimensions,
the best known upper bound is O(nk^(3/2)).
In its dual version, the problem can be generalized by replacing hyperplanes
by other families of surfaces (or curves in the planes). Reasonably sharp
bounds have been obtained for curves in the plane, but the known upper bounds
are rather weak for more general surfaces, already in three dimensions, except
for the case of triangles. The best known general bound, due to Chan is
O(n^2.997), for families of surfaces that satisfy certain (fairly weak)
properties.
In this paper we consider the case of pseudoplanes in 3 dimensions (defined
in detail in the introduction), and establish the upper bound O(nk^(5/3)) for
the number of k-level vertices in an arrangement of n pseudoplanes. The bound
is obtained by establishing suitable (and nontrivial) extensions of dual
versions of classical tools that have been used in studying the primal k-set
problem, such as the Lova'sz Lemma and the Crossing Lemma.Comment: 23 pages, 13 figure
Output-Sensitive Tools for Range Searching in Higher Dimensions
Let be a set of points in . A point is
\emph{-shallow} if it lies in a halfspace which contains at most points
of (including ). We show that if all points of are -shallow, then
can be partitioned into subsets, so that any hyperplane
crosses at most subsets. Given such
a partition, we can apply the standard construction of a spanning tree with
small crossing number within each subset, to obtain a spanning tree for the
point set , with crossing number . This allows us to extend the construction of Har-Peled
and Sharir \cite{hs11} to three and higher dimensions, to obtain, for any set
of points in (without the shallowness assumption), a
spanning tree with {\em small relative crossing number}. That is, any
hyperplane which contains points of on one side, crosses
edges of . Using a
similar mechanism, we also obtain a data structure for halfspace range
counting, which uses space (and somewhat higher
preprocessing cost), and answers a query in time , where is the output size
Selection Lemmas for various geometric objects
Selection lemmas are classical results in discrete geometry that have been
well studied and have applications in many geometric problems like weak epsilon
nets and slimming Delaunay triangulations. Selection lemma type results
typically show that there exists a point that is contained in many objects that
are induced (spanned) by an underlying point set.
In the first selection lemma, we consider the set of all the objects induced
(spanned) by a point set . This question has been widely explored for
simplices in , with tight bounds in . In our paper,
we prove first selection lemma for other classes of geometric objects. We also
consider the strong variant of this problem where we add the constraint that
the piercing point comes from . We prove an exact result on the strong and
the weak variant of the first selection lemma for axis-parallel rectangles,
special subclasses of axis-parallel rectangles like quadrants and slabs, disks
(for centrally symmetric point sets). We also show non-trivial bounds on the
first selection lemma for axis-parallel boxes and hyperspheres in
.
In the second selection lemma, we consider an arbitrary sized subset of
the set of all objects induced by . We study this problem for axis-parallel
rectangles and show that there exists an point in the plane that is contained
in rectangles. This is an improvement over the previous
bound by Smorodinsky and Sharir when is almost quadratic
Dense point sets have sparse Delaunay triangulations
The spread of a finite set of points is the ratio between the longest and
shortest pairwise distances. We prove that the Delaunay triangulation of any
set of n points in R^3 with spread D has complexity O(D^3). This bound is tight
in the worst case for all D = O(sqrt{n}). In particular, the Delaunay
triangulation of any dense point set has linear complexity. We also generalize
this upper bound to regular triangulations of k-ply systems of balls, unions of
several dense point sets, and uniform samples of smooth surfaces. On the other
hand, for any n and D=O(n), we construct a regular triangulation of complexity
Omega(nD) whose n vertices have spread D.Comment: 31 pages, 11 figures. Full version of SODA 2002 paper. Also available
at http://www.cs.uiuc.edu/~jeffe/pubs/screw.htm
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