共查询到20条相似文献,搜索用时 453 毫秒
1.
Given a collection S of subsets of some set
and
the set cover problem is to find the smallest subcollection
that covers
that is,
where
denotes
We assume of course that S covers
While the general problem is NP-hard to solve, even approximately, here we consider some geometric special cases, where usually
Combining previously known techniques [4], [5], we show that polynomial-time approximation algorithms with provable performance
exist, under a certain general condition: that for a random subset
and nondecreasing function f(·), there is a decomposition of the complement
into an expected at most f(|R|) regions, each region of a particular simple form. Under this condition, a cover of size O(f(|C|))
can be found in polynomial time. Using this result, and combinatorial geometry results implying bounding functions f(c) that
are nearly linear, we obtain o(log c) approximation algorithms for covering by fat triangles, by pseudo-disks, by a family
of fat objects, and others. Similarly, constant-factor approximations follow for similar-sized fat triangles and fat objects,
and for fat wedges. With more work, we obtain constant-factor approximation algorithms for covering by unit cubes in
and for guarding an x-monotone polygonal chain. 相似文献
2.
Daniel A. Klain 《Discrete and Computational Geometry》2006,36(3):457-477
Hyperbolic area is characterized as the unique continuous isometry-invariant simple valuation on convex polygons in
We then show that continuous isometry-invariant simple valuations on polytopes in
for
are determined uniquely by their values at ideal simplices. The proofs exploit a connection between valuation theory in
hyperbolic space and an analogous theory on the Euclidean sphere. These results lead to characterizations of continuous isometry-invariant
valuations on convex polytopes and convex bodies in the hyperbolic plane
a partial characterization in
and a mechanism for deriving many fundamental theorems of hyperbolic integral geometry, including kinematic formulas,
containment theorems, and isoperimetric and Bonnesen-type inequalities. 相似文献
3.
James H. Schmerl 《Discrete and Computational Geometry》2007,38(1):155-167
Suppose n ≥ 2 and
are such that X is infinite and Y is the set of vertices of an n-simplex. Then there is blue/red coloring of
such that no set similar to X is monochromatically blue and no set similar to Y is monochromatically red. 相似文献
4.
In this paper we study the worst-case error (of numerical integration) on the unit sphere
for all functions in the unit ball of the Sobolev space
where
More precisely, we consider infinite sequences
of m(n)-point numerical integration rules
where: (i)
is exact for all spherical polynomials of degree
and (ii)
has positive weights or, alternatively to (ii), the sequence
satisfies a certain local regularity property. Then we show that the worst-case error (of numerical integration)
in
has the upper bound
where the constant c depends on s and d (and possibly the sequence
This extends the recent results for the sphere
by K. Hesse and I.H. Sloan to spheres
of arbitrary dimension
by using an alternative representation of the worst-case error. If the sequence
of numerical integration rules satisfies
an order-optimal rate of convergence is achieved. 相似文献
5.
Rafal Latala Piotr Mankiewicz Krzysztof Oleszkiewicz Nicole Tomczak-Jaegermann 《Discrete and Computational Geometry》2007,38(1):29-50
We consider polytopes in
that are generated by N vectors in
whose coordinates are independent subgaussian random variables. (A particular case of such polytopes are symmetric random
polytopes generated by N independent vertices of the unit cube.) We show that for a random pair of such polytopes the Banach-Mazur
distance between them is essentially of a maximal order n. This result is an analogue of the well-known Gluskin's result
for spherical vectors. We also study the norms of projections on such polytopes and prove an analogue of Gluskin's and Szarek's
results on basis constants. The proofs are based on a version of "small ball" estimates for linear images of random subgaussian
vectors. 相似文献
6.
We give conditions on radial nonnegative weights $W_1We give conditions on radial nonnegative weights
and
on
, for which the a priori inequality
holds with constant independent of
. Here
is the Laplace-Beltrami operator on the sphere
. Due to the relation between
and the tangential component of the gradient,
, we obtain some "Morawetz-type" estimates for
on
. As a consequence we establish some new estimates for the free Schr?dinger propagator
, which may be viewed as certain refinements of the
-(super)smoothness estimates of Kato and Yajima. These results, in turn, lead to the well-posedness of the initial value problem
for certain time dependent first order spherical perturbations of the
dimensional Schr?dinger equation. 相似文献
7.
8.
Jesus Jeronimo Castro 《Discrete and Computational Geometry》2007,37(3):409-417
Let
be a family of convex figures in the plane. We say that
has property T if there exists a line intersecting every member of
. Also, the family
has property T(k) if every k-membered subfamily of
has property T. Let B be the unit disc centered at the origin. In this paper we prove that if a finite family
of translates of B has property T(4) then the family
, where
, has property T. We also give some results concerning families of translates of the unit disc which has either property T(3)
or property T(5). 相似文献
9.
An affine pseudo-plane X is a smooth affine surface defined over
which is endowed with an
-fibration such that every fiber is irreducible and only one fiber is a multiple fiber. If there is a hyperbolic
-action on X and X is an
-surface, we shall show that the universal covering
is isomorphic to an affine hypersurface
in the affine 3-space
and X is the quotient of
by the cyclic group
via the action
where
and
It is also shown that a
-homology plane X with
and a nontrivial
-action is an affine pseudo-plane. The automorphism group
is determined in the last section. 相似文献
10.
Given a finite subset
of an additive group
such as
or
, we are interested in efficient covering of
by translates of
, and efficient packing of translates of
in
. A set
provides a covering if the translates
with
cover
(i.e., their union is
), and the covering will be efficient if
has small density in
. On the other hand, a set
will provide a packing if the translated sets
with
are mutually disjoint, and the packing is efficient if
has large density.
In the present part (I) we will derive some facts on these concepts when
, and give estimates for the minimal covering densities and maximal packing densities of finite sets
. In part (II) we will again deal with
, and study the behaviour of such densities under linear transformations. In part (III) we will turn to
.
Authors’ address: Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, Colorado 80309-0395,
USA
The first author was partially supported by NSF DMS 0074531. 相似文献
11.
Zoltan Furedi 《Discrete and Computational Geometry》2007,38(2):273-288
Let
be a triangle and let
be a set of homothetic copies of
. We prove that
implies that there are positive and negative signs
and there exist translates of
that cover
. 相似文献
12.
Pedro J. Miana 《Semigroup Forum》2006,73(1):61-74
In this paper new equalities between two different convolution products in cancellative naturally ordered semigroups (but
not in groups) are given. We also give several applications in particular cases
and
相似文献
13.
Let
denote the linear space over
spanned by
. Define the (real) inner product
, where V satisfies: (i) V is real analytic on
; (ii)
; and (iii)
. Orthogonalisation of the (ordered) base
with respect to
yields the even degree and odd degree orthonormal Laurent polynomials
, and
. Define the even degree and odd degree monic orthogonal Laurent polynomials:
and
. Asymptotics in the double-scaling limit
such that
of
(in the entire complex plane),
, and
(in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a
matrix Riemann-Hilbert problem on
, and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further
developed in [2],[3]. 相似文献
14.
A compact set
is staircase connected if every two points
can be connected by a polygonal path with sides parallel to the coordinate axes, which is both x-monotone and y-monotone.
denotes the smallest number of edges of such a path.
is an integer-valued metric on S. We investigate this metric and introduce stars and kernels. Our main result is that the
r-th kernel is nonempty, compact and staircase connected provided
. 相似文献
15.
Frame Decomposition of Decomposition Spaces 总被引:3,自引:0,他引:3
A new construction of tight frames for
with flexible time-frequency localization
is considered. The frames can be adapted to form atomic decompositions for a large family of smoothness spaces on
a class of so-called decomposition spaces. The decomposition space norm can be completely characterized by a sparseness condition
on the frame coefficients. As examples of the general construction, new tight frames yielding decompositions of Besov space,
anisotropic Besov spaces, α-modulation spaces, and anisotropic α-modulation spaces are considered. Finally, curvelet-type
tight frames are constructed on
相似文献
16.
A triangulation of a set S of points in the plane is a subdivision of the convex hull of S into triangles whose vertices are
points of S. Given a set S of n points in
each moving independently, we wish to maintain a triangulation of S. The triangulation needs to be updated periodically as
the points in S move, so the goal is to maintain a triangulation with a small number of topological events, each being the
insertion or deletion of an edge. We propose a kinetic data structure (KDS) that processes
topological events with high probability if the trajectories of input points are algebraic curves of fixed degree. Each topological
event can be processed in
time. This is the first known KDS for maintaining a triangulation that processes a near-quadratic number of topological events,
and almost matches the
lower bound [1]. The number of topological events can be reduced to
if only k of the points are moving. 相似文献
17.
In this paper we show that there exists a
-coreset for k-median and k-means clustering of n points in
which is of size independent of n. In particular, we construct a
-coreset of size
for k-median clustering, and of size
for k-means clustering. 相似文献
18.
Given a function ψ in
the affine (wavelet) system generated by ψ, associated to an invertible matrix a and a lattice Γ, is the collection of functions
In this paper we prove that the set of functions generating affine systems that are a Riesz basis of
${\cal L}^2({\Bbb R}^d)$ is dense in We also prove that a stronger result is true for affine systems that are a frame of
In this case we show that the generators associated to a fixed but arbitrary dilation are a dense set. Furthermore, we analyze
the orthogonal case in which we prove that the set of generators of orthogonal (not necessarily complete) affine systems,
that are compactly supported in frequency, are dense in the unit sphere of
with the induced metric. As a byproduct we introduce the p-Grammian of a function and prove a convergence result of this
Grammian as a function of the lattice. This result gives insight in the problem of oversampling of affine systems. 相似文献
19.
António M. Caetano Hans-Gerd Leopold 《Journal of Fourier Analysis and Applications》2006,12(4):427-445
The concept of local growth envelope
of the quasi-normed function space
is applied to the Triebel-Lizorkin spaces of generalized smoothness
In order to achieve this, a standardization result for these and corresponding Besov spaces is derived. 相似文献
20.
On the Least Median Square Problem 总被引:1,自引:0,他引:1
Jeff Erickson Sariel Har-Peled David M. Mount 《Discrete and Computational Geometry》2006,36(4):593-607
We consider the exact and approximate computational complexity of the multivariate least median-of-squares (LMS) linear regression
estimator. The LMS estimator is among the most widely used robust linear statistical estimators. Given a set of n points in
and a parameter k, the problem is equivalent to computing the narrowest slab bounded by two parallel hyperplanes that contains
k of the points. We present algorithms for the exact and approximate versions of the multivariate LMS problem. We also provide
nearly matching lower bounds for these problems. These lower bounds hold under the assumptions that k is Ω(n) and that deciding
whether n given points in
are affinely non-degenerate requires Ω(nd) time. 相似文献