4,059 research outputs found
Net and Prune: A Linear Time Algorithm for Euclidean Distance Problems
We provide a general framework for getting expected linear time constant
factor approximations (and in many cases FPTAS's) to several well known
problems in Computational Geometry, such as -center clustering and farthest
nearest neighbor. The new approach is robust to variations in the input
problem, and yet it is simple, elegant and practical. In particular, many of
these well studied problems which fit easily into our framework, either
previously had no linear time approximation algorithm, or required rather
involved algorithms and analysis. A short list of the problems we consider
include farthest nearest neighbor, -center clustering, smallest disk
enclosing points, th largest distance, th smallest -nearest
neighbor distance, th heaviest edge in the MST and other spanning forest
type problems, problems involving upward closed set systems, and more. Finally,
we show how to extend our framework such that the linear running time bound
holds with high probability
From Proximity to Utility: A Voronoi Partition of Pareto Optima
We present an extension of Voronoi diagrams where when considering which site
a client is going to use, in addition to the site distances, other site
attributes are also considered (for example, prices or weights). A cell in this
diagram is then the locus of all clients that consider the same set of sites to
be relevant. In particular, the precise site a client might use from this
candidate set depends on parameters that might change between usages, and the
candidate set lists all of the relevant sites. The resulting diagram is
significantly more expressive than Voronoi diagrams, but naturally has the
drawback that its complexity, even in the plane, might be quite high.
Nevertheless, we show that if the attributes of the sites are drawn from the
same distribution (note that the locations are fixed), then the expected
complexity of the candidate diagram is near linear.
To this end, we derive several new technical results, which are of
independent interest. In particular, we provide a high-probability,
asymptotically optimal bound on the number of Pareto optima points in a point
set uniformly sampled from the -dimensional hypercube. To do so we revisit
the classical backward analysis technique, both simplifying and improving
relevant results in order to achieve the high-probability bounds
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
Tunable stacking fault energies by tailoring local chemical order in CrCoNi medium-entropy alloys
High-entropy alloys (HEAs) are an intriguing new class of metallic materials
due to their unique mechanical behavior. Achieving a detailed understanding of
structure-property relationships in these materials has been challenged by the
compositional disorder that underlies their unique mechanical behavior.
Accordingly, in this work, we employ first-principles calculations to
investigate the nature of local chemical order and establish its relationship
to the intrinsic and extrinsic stacking fault energy (SFE) in CrCoNi
medium-entropy solid-solution alloys, whose combination of strength, ductility
and toughness properties approach the best on record. We find that the average
intrinsic and extrinsic SFE are both highly tunable, with values ranging from
-43 mJ.m-2 to 30 mJ.m-2 and from -28 mJ.m-2 to 66 mJ.m-2, respectively, as the
degree of local chemical order increases. The state of local ordering also
strongly correlates with the energy difference between the face-centered cubic
(fcc) and hexagonal-close packed (hcp) phases, which affects the occurrence of
transformation-induced plasticity. This theoretical study demonstrates that
chemical short-range order is thermodynamically favored in HEAs and can be
tuned to affect the mechanical behavior of these alloys. It thus addresses the
pressing need to establish robust processing-structure-property relationships
to guide the science-based design of new HEAs with targeted mechanical
behavior.Comment: 23 pages, 5 figure
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