847 research outputs found
Down the Rabbit Hole: Robust Proximity Search and Density Estimation in Sublinear Space
For a set of points in , and parameters and \eps, we present
a data structure that answers (1+\eps,k)-\ANN queries in logarithmic time.
Surprisingly, the space used by the data-structure is \Otilde (n /k); that
is, the space used is sublinear in the input size if is sufficiently large.
Our approach provides a novel way to summarize geometric data, such that
meaningful proximity queries on the data can be carried out using this sketch.
Using this, we provide a sublinear space data-structure that can estimate the
density of a point set under various measures, including:
\begin{inparaenum}[(i)]
\item sum of distances of closest points to the query point, and
\item sum of squared distances of closest points to the query point.
\end{inparaenum}
Our approach generalizes to other distance based estimation of densities of
similar flavor. We also study the problem of approximating some of these
quantities when using sampling. In particular, we show that a sample of size
\Otilde (n /k) is sufficient, in some restricted cases, to estimate the above
quantities. Remarkably, the sample size has only linear dependency on the
dimension
Searchable Sky Coverage of Astronomical Observations: Footprints and Exposures
Sky coverage is one of the most important pieces of information about
astronomical observations. We discuss possible representations, and present
algorithms to create and manipulate shapes consisting of generalized spherical
polygons with arbitrary complexity and size on the celestial sphere. This shape
specification integrates well with our Hierarchical Triangular Mesh indexing
toolbox, whose performance and capabilities are enhanced by the advanced
features presented here. Our portable implementation of the relevant spherical
geometry routines comes with wrapper functions for database queries, which are
currently being used within several scientific catalog archives including the
Sloan Digital Sky Survey, the Galaxy Evolution Explorer and the Hubble Legacy
Archive projects as well as the Footprint Service of the Virtual Observatory.Comment: 11 pages, 7 figures, submitted to PAS
Spanners for Geometric Intersection Graphs
Efficient algorithms are presented for constructing spanners in geometric
intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is
obtained using efficient partitioning of the space into hypercubes and solving
bichromatic closest pair problems. The spanner construction has almost
equivalent complexity to the construction of Euclidean minimum spanning trees.
The results are extended to arbitrary ball graphs with a sub-quadratic running
time.
For unit ball graphs, the spanners have a small separator decomposition which
can be used to obtain efficient algorithms for approximating proximity problems
like diameter and distance queries. The results on compressed quadtrees,
geometric graph separators, and diameter approximation might be of independent
interest.Comment: 16 pages, 5 figures, Late
Width and mode of the profile for some random trees of logarithmic height
We propose a new, direct, correlation-free approach based on central moments
of profiles to the asymptotics of width (size of the most abundant level) in
some random trees of logarithmic height. The approach is simple but gives
precise estimates for expected width, central moments of the width and almost
sure convergence. It is widely applicable to random trees of logarithmic
height, including recursive trees, binary search trees, quad trees,
plane-oriented ordered trees and other varieties of increasing trees.Comment: Published at http://dx.doi.org/10.1214/105051606000000187 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
The Representation of symmetric patterns using the quadtree data structure
Hierarchical data structures for image representation have been widely explored in recent years. These data structures are based on the principle of recursive decomposition of an image region. The most commonly mentioned picture data structure for two-dimensional data is referred to as a quadtree . The purpose of this thesis is to investigate the use of a general quadtree scheme as a means of representing symmetric images. Specifically, images are generated according to the rules of selected two-dimensional plane symmetry groups
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