76,327 research outputs found
Onion Curve: A Space Filling Curve with Near-Optimal Clustering
Space filling curves (SFCs) are widely used in the design of indexes for
spatial and temporal data. Clustering is a key metric for an SFC, that measures
how well the curve preserves locality in moving from higher dimensions to a
single dimension. We present the {\em onion curve}, an SFC whose clustering
performance is provably close to optimal for the cube and near-cube shaped
query sets, irrespective of the side length of the query. We show that in
contrast, the clustering performance of the widely used Hilbert curve can be
far from optimal, even for cube-shaped queries. Since the clustering
performance of an SFC is critical to the efficiency of multi-dimensional
indexes based on the SFC, the onion curve can deliver improved performance for
data structures involving multi-dimensional data.Comment: The short version is published in ICDE 1
Effectiveness of landmark analysis for establishing locality in p2p networks
Locality to other nodes on a peer-to-peer overlay network can be established by means of a set of landmarks shared among the participating nodes. Each node independently collects a set of latency measures to landmark nodes, which are used as a multi-dimensional feature vector. Each peer node uses the feature vector to generate a unique scalar index which is correlated to its topological locality. A popular dimensionality reduction technique is the space filling Hilbert’s curve, as it possesses good locality
preserving properties. However, there exists little comparison between Hilbert’s curve and other techniques for dimensionality reduction. This work carries out a quantitative analysis of their properties. Linear and non-linear techniques for scaling the landmark vectors to a single dimension are investigated. Hilbert’s curve, Sammon’s mapping and Principal Component Analysis
have been used to generate a 1d space with locality preserving properties. This work provides empirical evidence to support the use of Hilbert’s curve in the context of locality preservation when generating peer identifiers by means of landmark vector analysis. A comparative analysis is carried out with an artificial 2d network model and with a realistic network topology model
with a typical power-law distribution of node connectivity in the Internet. Nearest neighbour analysis confirms Hilbert’s curve to be very effective in both artificial and realistic network topologies. Nevertheless, the results in the realistic network model show that there is scope for improvements and better techniques to preserve locality information are required
Deterministic global optimization using space-filling curves and multiple estimates of Lipschitz and Holder constants
In this paper, the global optimization problem with
being a hyperinterval in and satisfying the Lipschitz condition
with an unknown Lipschitz constant is considered. It is supposed that the
function can be multiextremal, non-differentiable, and given as a
`black-box'. To attack the problem, a new global optimization algorithm based
on the following two ideas is proposed and studied both theoretically and
numerically. First, the new algorithm uses numerical approximations to
space-filling curves to reduce the original Lipschitz multi-dimensional problem
to a univariate one satisfying the H\"{o}lder condition. Second, the algorithm
at each iteration applies a new geometric technique working with a number of
possible H\"{o}lder constants chosen from a set of values varying from zero to
infinity showing so that ideas introduced in a popular DIRECT method can be
used in the H\"{o}lder global optimization. Convergence conditions of the
resulting deterministic global optimization method are established. Numerical
experiments carried out on several hundreds of test functions show quite a
promising performance of the new algorithm in comparison with its direct
competitors.Comment: 26 pages, 10 figures, 4 table
Superfluidity of fermions with repulsive on-site interaction in an anisotropic optical lattice near a Feshbach resonance
We present a numerical study on ground state properties of a one-dimensional
(1D) general Hubbard model (GHM) with particle-assisted tunnelling rates and
repulsive on-site interaction (positive-U), which describes fermionic atoms in
an anisotropic optical lattice near a wide Feshbach resonance. For our
calculation, we utilize the time evolving block decimation (TEBD) algorithm,
which is an extension of the density matrix renormalization group and provides
a well-controlled method for 1D systems. We show that the positive-U GHM, when
hole-doped from half-filling, exhibits a phase with coexistence of
quasi-long-range superfluid and charge-density-wave orders. This feature is
different from the property of the conventional Hubbard model with positive-U,
indicating the particle-assisted tunnelling mechanism in GHM brings in
qualitatively new physics.Comment: updated with published version
Index Information Algorithm with Local Tuning for Solving Multidimensional Global Optimization Problems with Multiextremal Constraints
Multidimensional optimization problems where the objective function and the
constraints are multiextremal non-differentiable Lipschitz functions (with
unknown Lipschitz constants) and the feasible region is a finite collection of
robust nonconvex subregions are considered. Both the objective function and the
constraints may be partially defined. To solve such problems an algorithm is
proposed, that uses Peano space-filling curves and the index scheme to reduce
the original problem to a H\"{o}lder one-dimensional one. Local tuning on the
behaviour of the objective function and constraints is used during the work of
the global optimization procedure in order to accelerate the search. The method
neither uses penalty coefficients nor additional variables. Convergence
conditions are established. Numerical experiments confirm the good performance
of the technique.Comment: 29 pages, 5 figure
Sixteen space-filling curves and traversals for d-dimensional cubes and simplices
This article describes sixteen different ways to traverse d-dimensional space
recursively in a way that is well-defined for any number of dimensions. Each of
these traversals has distinct properties that may be beneficial for certain
applications. Some of the traversals are novel, some have been known in
principle but had not been described adequately for any number of dimensions,
some of the traversals have been known. This article is the first to present
them all in a consistent notation system. Furthermore, with this article, tools
are provided to enumerate points in a regular grid in the order in which they
are visited by each traversal. In particular, we cover: five discontinuous
traversals based on subdividing cubes into 2^d subcubes: Z-traversal (Morton
indexing), U-traversal, Gray-code traversal, Double-Gray-code traversal, and
Inside-out traversal; two discontinuous traversals based on subdividing
simplices into 2^d subsimplices: the Hill-Z traversal and the Maehara-reflected
traversal; five continuous traversals based on subdividing cubes into 2^d
subcubes: the Base-camp Hilbert curve, the Harmonious Hilbert curve, the Alfa
Hilbert curve, the Beta Hilbert curve, and the Butz-Hilbert curve; four
continuous traversals based on subdividing cubes into 3^d subcubes: the Peano
curve, the Coil curve, the Half-coil curve, and the Meurthe curve. All of these
traversals are self-similar in the sense that the traversal in each of the
subcubes or subsimplices of a cube or simplex, on any level of recursive
subdivision, can be obtained by scaling, translating, rotating, reflecting
and/or reversing the traversal of the complete unit cube or simplex.Comment: 28 pages, 12 figures. v2: fixed a confusing typo on page 12, line
Discrete denoising of heterogenous two-dimensional data
We consider discrete denoising of two-dimensional data with characteristics
that may be varying abruptly between regions.
Using a quadtree decomposition technique and space-filling curves, we extend
the recently developed S-DUDE (Shifting Discrete Universal DEnoiser), which was
tailored to one-dimensional data, to the two-dimensional case. Our scheme
competes with a genie that has access, in addition to the noisy data, also to
the underlying noiseless data, and can employ different two-dimensional
sliding window denoisers along distinct regions obtained by a quadtree
decomposition with leaves, in a way that minimizes the overall loss. We
show that, regardless of what the underlying noiseless data may be, the
two-dimensional S-DUDE performs essentially as well as this genie, provided
that the number of distinct regions satisfies , where is the total
size of the data. The resulting algorithm complexity is still linear in both
and , as in the one-dimensional case. Our experimental results show that
the two-dimensional S-DUDE can be effective when the characteristics of the
underlying clean image vary across different regions in the data.Comment: 16 pages, submitted to IEEE Transactions on Information Theor
A Study of Energy and Locality Effects using Space-filling Curves
The cost of energy is becoming an increasingly important driver for the
operating cost of HPC systems, adding yet another facet to the challenge of
producing efficient code. In this paper, we investigate the energy implications
of trading computation for locality using Hilbert and Morton space-filling
curves with dense matrix-matrix multiplication. The advantage of these curves
is that they exhibit an inherent tiling effect without requiring specific
architecture tuning. By accessing the matrices in the order determined by the
space-filling curves, we can trade computation for locality. The index
computation overhead of the Morton curve is found to be balanced against its
locality and energy efficiency, while the overhead of the Hilbert curve
outweighs its improvements on our test system.Comment: Proceedings of the 2014 IEEE International Parallel & Distributed
Processing Symposium Workshops (IPDPSW
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