81,349 research outputs found
Recursive tilings and space-filling curves with little fragmentation
This paper defines the Arrwwid number of a recursive tiling (or space-filling
curve) as the smallest number w such that any ball Q can be covered by w tiles
(or curve sections) with total volume O(vol(Q)). Recursive tilings and
space-filling curves with low Arrwwid numbers can be applied to optimise disk,
memory or server access patterns when processing sets of points in
d-dimensional space. This paper presents recursive tilings and space-filling
curves with optimal Arrwwid numbers. For d >= 3, we see that regular cube
tilings and space-filling curves cannot have optimal Arrwwid number, and we see
how to construct alternatives with better Arrwwid numbers.Comment: Manuscript accompanying abstract in EuroCG 2010, including full
proofs, 20 figures, references, discussion et
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
Neural Space-filling Curves
We present Neural Space-filling Curves (SFCs), a data-driven approach to
infer a context-based scan order for a set of images. Linear ordering of pixels
forms the basis for many applications such as video scrambling, compression,
and auto-regressive models that are used in generative modeling for images.
Existing algorithms resort to a fixed scanning algorithm such as Raster scan or
Hilbert scan. Instead, our work learns a spatially coherent linear ordering of
pixels from the dataset of images using a graph-based neural network. The
resulting Neural SFC is optimized for an objective suitable for the downstream
task when the image is traversed along with the scan line order. We show the
advantage of using Neural SFCs in downstream applications such as image
compression. Code and additional results will be made available at
https://hywang66.github.io/publication/neuralsfc
Locality and Bounding-Box Quality of Two-Dimensional Space-Filling Curves
Space-filling curves can be used to organise points in the plane into
bounding-box hierarchies (such as R-trees). We develop measures of the
bounding-box quality of space-filling curves that express how effective
different space-filling curves are for this purpose. We give general lower
bounds on the bounding-box quality measures and on locality according to
Gotsman and Lindenbaum for a large class of space-filling curves. We describe a
generic algorithm to approximate these and similar quality measures for any
given curve. Using our algorithm we find good approximations of the locality
and the bounding-box quality of several known and new space-filling curves.
Surprisingly, some curves with relatively bad locality by Gotsman and
Lindenbaum's measure, have good bounding-box quality, while the curve with the
best-known locality has relatively bad bounding-box quality.Comment: 24 pages, full version of paper to appear in ESA. Difference with
first version: minor editing; Fig. 2(m) correcte
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