86,162 research outputs found
Estimating Multidimensional Persistent Homology through a Finite Sampling
An exact computation of the persistent Betti numbers of a submanifold of
a Euclidean space is possible only in a theoretical setting. In practical
situations, only a finite sample of is available. We show that, under
suitable density conditions, it is possible to estimate the multidimensional
persistent Betti numbers of from the ones of a union of balls centered on
the sample points; this even yields the exact value in restricted areas of the
domain.
Using these inequalities we improve a previous lower bound for the natural
pseudodistance to assess dissimilarity between the shapes of two objects from a
sampling of them.
Similar inequalities are proved for the multidimensional persistent Betti
numbers of the ball union and the one of a combinatorial description of it
The Skip Quadtree: A Simple Dynamic Data Structure for Multidimensional Data
We present a new multi-dimensional data structure, which we call the skip
quadtree (for point data in R^2) or the skip octree (for point data in R^d,
with constant d>2). Our data structure combines the best features of two
well-known data structures, in that it has the well-defined "box"-shaped
regions of region quadtrees and the logarithmic-height search and update
hierarchical structure of skip lists. Indeed, the bottom level of our structure
is exactly a region quadtree (or octree for higher dimensional data). We
describe efficient algorithms for inserting and deleting points in a skip
quadtree, as well as fast methods for performing point location and approximate
range queries.Comment: 12 pages, 3 figures. A preliminary version of this paper appeared in
the 21st ACM Symp. Comp. Geom., Pisa, 2005, pp. 296-30
Partial-sum queries in OLAP data cubes using covering codes
A partial-sum query obtains the summation over a set of specified cells of a data cube. We establish a connection between the covering problem in the theory of error-correcting codes and the partial-sum problem and use this connection to devise algorithms for the partial-sum problem with efficient space-time trade-offs. For example, using our algorithms, with 44 percent additional storage, the query response time can be improved by about 12 percent; by roughly doubling the storage requirement, the query response time can be improved by about 34 percent
Some Problems of Topology Change Description in the Theory of Space-Time
The problem of topology change description in gravitation theory is analized
in detailes. It is pointed out that in standard four-dimensional theories the
topology of space may be considered as a particular case of boundary conditions
(or constraints). Therefore, the possible changes of space topology in
(3+1)-dimensions do not admit dynamical description nor in classical nor in
quantum theories and the statements about dynamical supressing of topology
change have no sence. In the framework of multidimensional theories the space
(and space-time) may be considered as the embedded manifolds. It give the real
posibilities for the dynamical description of the topology of space or
space-time.Comment: 16 pages, LaTex; some misprints, which prevent generation of PS file,
are correcte
Continuous Fields and Discrete Samples: Reconstruction through Delaunay Tessellations
Here we introduce the Delaunay Density Estimator Method. Its purpose is
rendering a fully volume-covering reconstruction of a density field from a set
of discrete data points sampling this field. Reconstructing density or
intensity fields from a set of irregularly sampled data is a recurring key
issue in operations on astronomical data sets, both in an observational context
as well as in the context of numerical simulations. Our technique is based upon
the stochastic geometric concept of the Delaunay tessellation generated by the
point set. We shortly describe the method, and illustrate its virtues by means
of an application to an N-body simulation of cosmic structure formation. The
presented technique is a fully adaptive method: automatically it probes high
density regions at maximum possible resolution, while low density regions are
recovered as moderately varying regions devoid of the often irritating
shot-noise effects. Of equal importance is its capability to sharply and
undilutedly recover anisotropic density features like filaments and walls. The
prominence of such features at a range of resolution levels within a
hierarchical clustering scenario as the example of the standard CDM scenario is
shown to be impressively recovered by our scheme.Comment: 4 pages, 2 figures, accepted for publication in Astronomy &
Astrophysics Letter
Near-Optimal and Robust Mechanism Design for Covering Problems with Correlated Players
We consider the problem of designing incentive-compatible, ex-post
individually rational (IR) mechanisms for covering problems in the Bayesian
setting, where players' types are drawn from an underlying distribution and may
be correlated, and the goal is to minimize the expected total payment made by
the mechanism. We formulate a notion of incentive compatibility (IC) that we
call {\em support-based IC} that is substantially more robust than Bayesian IC,
and develop black-box reductions from support-based-IC mechanism design to
algorithm design. For single-dimensional settings, this black-box reduction
applies even when we only have an LP-relative {\em approximation algorithm} for
the algorithmic problem. Thus, we obtain near-optimal mechanisms for various
covering settings including single-dimensional covering problems, multi-item
procurement auctions, and multidimensional facility location.Comment: Major changes compared to the previous version. Please consult this
versio
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