6 research outputs found
Hyperorthogonal well-folded Hilbert curves
R-trees can be used to store and query sets of point data in two or more
dimensions. An easy way to construct and maintain R-trees for two-dimensional
points, due to Kamel and Faloutsos, is to keep the points in the order in which
they appear along the Hilbert curve. The R-tree will then store bounding boxes
of points along contiguous sections of the curve, and the efficiency of the
R-tree depends on the size of the bounding boxes---smaller is better. Since
there are many different ways to generalize the Hilbert curve to higher
dimensions, this raises the question which generalization results in the
smallest bounding boxes. Familiar methods, such as the one by Butz, can result
in curve sections whose bounding boxes are a factor larger
than the volume traversed by that section of the curve. Most of the volume
bounded by such bounding boxes would not contain any data points. In this paper
we present a new way of generalizing Hilbert's curve to higher dimensions,
which results in much tighter bounding boxes: they have at most 4 times the
volume of the part of the curve covered, independent of the number of
dimensions. Moreover, we prove that a factor 4 is asymptotically optimal.Comment: Manuscript submitted to Journal of Computational Geometry. An
abstract appeared in the 31st Int Symp on Computational Geometry (SoCG 2015),
LIPIcs 34:812-82
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
Computation of Lebesgue’s Space-Filling Curve
The means of realizing or approximating the Lebesgue space-filling curve (SFC) with binary arithmetic on a uniformly spaced binary grid are not obvious, one problem being its formulation in terms of ternary representations; that impediment can be overcome via use of a binary-oriented Cantor set. A second impediment, namely the Devil’s Staircase feature, also created by the role of the Cantor set, can be overcome via the definition of a “working inverse”, thereby providing means of achieving compatibility with such a grid. The results indicate an alternative way to proceed, in realizing an approximation to Lebesgue’s SFC, which circumvents any complication raised by Cantor sets and is compatible with binary and integer arithmetic. Well-known constructions such as the z-curve or Morton order, sometimes considered in association with Lebesgue’s SFC, are treated as irrelevant
Hyperorthogonal well-folded Hilbert curves HYPERORTHOGONAL WELL-FOLDED HILBERT CURVES
Abstract R-trees can be used to store and query sets of point data in two or more dimensions. An easy way to construct and maintain R-trees for two-dimensional points, due to Kamel and Faloutsos, is to keep the points in the order in which they appear along the Hilbert curve. The R-tree will then store bounding boxes of points along contiguous sections of the curve, and the efficiency of the R-tree depends on the size of the bounding boxes-smaller is better. Since there are many different ways to generalize the Hilbert curve to higher dimensions, this raises the question which generalization results in the smallest bounding boxes. Familiar methods, such as the one by Butz, can result in curve sections whose bounding boxes are a factor Ω(2 d/2 ) larger than the volume traversed by that section of the curve. Most of the volume bounded by such bounding boxes would not contain any data points. In this paper we present a new way of generalizing Hilbert's curve to higher dimensions, which results in much tighter bounding boxes: they have at most 4 times the volume of the part of the curve covered, independent of the number of dimensions. Moreover, we prove that a factor 4 is asymptotically optimal
Hyperorthogonal well-folded Hilbert curves
R-trees can be used to store and query sets of point data in two or more dimensions. An easy way to construct and maintain R-trees for two-dimensional points, due to Kamel and Faloutsos, is to keep the points in the order in which they appear along the Hilbert curve. The R-tree will then store bounding boxes of points along contiguous sections of the curve, and the efficiency of the R-tree depends on the size of the bounding boxes---smaller is better. Since there are many different ways to generalize the Hilbert curve to higher dimensions, this raises the question which generalization results in the smallest bounding boxes. Familiar methods, such as the one by Butz, can result in curve sections whose bounding boxes are a factor Ω(2 d/2 ) \u3cbr/\u3eΩ(2d/2)\u3cbr/\u3e larger than the volume traversed by that section of the curve. Most of the volume bounded by such bounding boxes would not contain any data points. In this paper we present a new way of generalizing Hilbert's curve to higher dimensions, which results in much tighter bounding boxes: they have at most 4 times the volume of the part of the curve covered, independent of the number of dimensions. Moreover, we prove that a factor 4 is asymptotically optimal