91,947 research outputs found
Multidimensional Range Queries on Modern Hardware
Range queries over multidimensional data are an important part of database
workloads in many applications. Their execution may be accelerated by using
multidimensional index structures (MDIS), such as kd-trees or R-trees. As for
most index structures, the usefulness of this approach depends on the
selectivity of the queries, and common wisdom told that a simple scan beats
MDIS for queries accessing more than 15%-20% of a dataset. However, this wisdom
is largely based on evaluations that are almost two decades old, performed on
data being held on disks, applying IO-optimized data structures, and using
single-core systems. The question is whether this rule of thumb still holds
when multidimensional range queries (MDRQ) are performed on modern
architectures with large main memories holding all data, multi-core CPUs and
data-parallel instruction sets. In this paper, we study the question whether
and how much modern hardware influences the performance ratio between index
structures and scans for MDRQ. To this end, we conservatively adapted three
popular MDIS, namely the R*-tree, the kd-tree, and the VA-file, to exploit
features of modern servers and compared their performance to different flavors
of parallel scans using multiple (synthetic and real-world) analytical
workloads over multiple (synthetic and real-world) datasets of varying size,
dimensionality, and skew. We find that all approaches benefit considerably from
using main memory and parallelization, yet to varying degrees. Our evaluation
indicates that, on current machines, scanning should be favored over parallel
versions of classical MDIS even for very selective queries
Incidences between points and lines in three dimensions
We give a fairly elementary and simple proof that shows that the number of
incidences between points and lines in , so that no
plane contains more than lines, is (in the precise statement, the constant
of proportionality of the first and third terms depends, in a rather weak
manner, on the relation between and ).
This bound, originally obtained by Guth and Katz~\cite{GK2} as a major step
in their solution of Erd{\H o}s's distinct distances problem, is also a major
new result in incidence geometry, an area that has picked up considerable
momentum in the past six years. Its original proof uses fairly involved
machinery from algebraic and differential geometry, so it is highly desirable
to simplify the proof, in the interest of better understanding the geometric
structure of the problem, and providing new tools for tackling similar
problems. This has recently been undertaken by Guth~\cite{Gu14}. The present
paper presents a different and simpler derivation, with better bounds than
those in \cite{Gu14}, and without the restrictive assumptions made there. Our
result has a potential for applications to other incidence problems in higher
dimensions
On Range Searching with Semialgebraic Sets II
Let be a set of points in . We present a linear-size data
structure for answering range queries on with constant-complexity
semialgebraic sets as ranges, in time close to . It essentially
matches the performance of similar structures for simplex range searching, and,
for , significantly improves earlier solutions by the first two authors
obtained in~1994. This almost settles a long-standing open problem in range
searching.
The data structure is based on the polynomial-partitioning technique of Guth
and Katz [arXiv:1011.4105], which shows that for a parameter , , there exists a -variate polynomial of degree such that
each connected component of contains at most points
of , where is the zero set of . We present an efficient randomized
algorithm for computing such a polynomial partition, which is of independent
interest and is likely to have additional applications
Bandwidth selection for kernel estimation in mixed multi-dimensional spaces
Kernel estimation techniques, such as mean shift, suffer from one major
drawback: the kernel bandwidth selection. The bandwidth can be fixed for all
the data set or can vary at each points. Automatic bandwidth selection becomes
a real challenge in case of multidimensional heterogeneous features. This paper
presents a solution to this problem. It is an extension of \cite{Comaniciu03a}
which was based on the fundamental property of normal distributions regarding
the bias of the normalized density gradient. The selection is done iteratively
for each type of features, by looking for the stability of local bandwidth
estimates across a predefined range of bandwidths. A pseudo balloon mean shift
filtering and partitioning are introduced. The validity of the method is
demonstrated in the context of color image segmentation based on a
5-dimensional space
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