Table of Contents
Introduction
This is a package to convert between one dimensional distance along a
Hilbert curve, h, and n-dimensional points,
(x_0, x_1, ... x_n-1). There are two important parameters,
n-- the number of dimensions (must be > 0)p-- the number of iterations used in constructing the Hilbert curve (must be > 0)
We consider an n-dimensional hypercube of side length 2^p.
This hypercube contains 2^{n p} unit hypercubes (2^p along
each dimension). The number of unit hypercubes determine the possible
discrete distances along the Hilbert curve (indexed from 0 to
2^{n p} - 1).
Quickstart
Install the package with pip,
pip install hilbertcurveYou can calculate points given distances along a hilbert curve,
>>> from hilbertcurve.hilbertcurve import HilbertCurve
>>> p=1; n=2
>>> hilbert_curve = HilbertCurve(p, n)
>>> distances = list(range(4))
>>> points = hilbert_curve.points_from_distances(distances)
>>> for point, dist in zip(points, distances):
>>> print(f'point(h={dist}) = {point}')
point(h=0) = [0, 0]
point(h=1) = [0, 1]
point(h=2) = [1, 1]
point(h=3) = [1, 0]You can also calculate distances along a hilbert curve given points,
>>> points = [[0,0], [0,1], [1,1], [1,0]]
>>> distances = hilbert_curve.distances_from_points(points)
>>> for point, dist in zip(points, distances):
>>> print(f'distance(x={point}) = {dist}')
distance(x=[0, 0]) = 0
distance(x=[0, 1]) = 1
distance(x=[1, 1]) = 2
distance(x=[1, 0]) = 3Inputs and Outputs
The HilbertCurve class has four main public methods.
point_from_distance(distance: int) -> Iterable[int]points_from_distances(distances: Iterable[int], match_type: bool=False) -> Iterable[Iterable[int]]distance_from_point(point: Iterable[int]) -> intdistances_from_points(points: Iterable[Iterable[int]], match_type: bool=False) -> Iterable[int]
Arguments that are type hinted with Iterable[int] have been tested with lists, tuples, and 1-d numpy arrays.
Arguments that are type hinted with Iterable[Iterable[int]] have been tested with list of lists, tuples of tuples, and 2-d numpy arrays with shape (num_points, num_dimensions). The match_type key word argument forces the output iterable to match the type of the input iterable.
The HilbertCurve class also contains some useful metadata derived from the inputs p and n. For instance, you can construct a numpy array of random points on the hilbert curve and calculate their distances in the following way,
>>> from hilbertcurve.hilbertcurve import HilbertCurve
>>> p=1; n=2
>>> hilbert_curve = HilbertCurve(p, n)
>>> num_points = 10_000
>>> points = np.random.randint(
low=0,
high=hilbert_curve.max_x + 1,
size=(num_points, hilbert_curve.n)
)
>>> distances = hilbert_curve.distances_from_points(points)
>>> type(distances)
list
>>> distances = hilbert_curve.distances_from_points(points, match_type=True)
>>> type(distances)
numpy.ndarrayMultiprocessing
You can now take advantage of multiple processes to speed up calculations by using the n_procs keyword argument when creating an instance of HilbertCurve.
n_procs (int): number of processes to use
0 = dont use multiprocessing
-1 = use all available processes
any other positive integer = number of processes to useA value of 0 will completely avoid using the multiprocessing module while a value of 1 will use the multiprocessing module but with a single process. If you want to take advantage of every thread on your computer use the value -1 and if you want something in the middle use a value between 1 and the number of threads on your computer. A concrete example starting with the code block above is,
>>> from hilbertcurve.hilbertcurve import HilbertCurve
>>> p=1; n=2
>>> hilbert_curve = HilbertCurve(p, n, n_procs=-1)
>>> num_points = 100_000
>>> points = np.random.randint(
low=0,
high=hilbert_curve.max_x + 1,
size=(num_points, hilbert_curve.n)
)
>>> distances = hilbert_curve.distances_from_points(points)The following methods are able to use multiple cores.
points_from_distances(distances: Iterable[int], match_type: bool=False) -> Iterable[Iterable[int]]distances_from_points(points: Iterable[Iterable[int]], match_type: bool=False) -> Iterable[int]
(Absurdly) Large Integers
Due to the magic of arbitrarily large integers in Python, these calculations can be done with ... well ... arbitrarily large integers!
>>> p = 512; n = 10
>>> hilbert_curve = HilbertCurve(p, n)
>>> ii = 123456789101112131415161718192021222324252627282930
>>> point = hilbert_curve.points_from_distances([ii])[0]
>>> print(f'point = {point}')
point = [121075, 67332, 67326, 108879, 26637, 43346, 23848, 1551, 68130, 84004]The calculations above represent the 512th iteration of the Hilbert curve in 10 dimensions.
The maximum value along any coordinate axis is an integer with 155 digits and the maximum
distance along the curve is an integer with 1542 digits. For comparison,
an estimate of the number of atoms in the observable universe
is 10^{82} (i.e. an integer with 83 digits).
Visuals
The figure above shows the first three iterations of the Hilbert
curve in two (n=2) dimensions. The p=1 iteration is shown
in red, p=2 in blue, and p=3 in black.
For the p=3 iteration, distances, h, along the curve are
labeled from 0 to 63 (i.e. from 0 to 2^{n p}-1). This package
provides methods to translate between n-dimensional points and one
dimensional distance. For example, between (x_0=4, x_1=6) and
h=36. Note that the p=1 and p=2 iterations have been
scaled and translated to the coordinate system of the p=3 iteration.
An animation of the same case in 3-D is available on YouTube. To watch the video, click the link below. Once the YouTube video loads, you can right click on it and turn "Loop" on to watch the curve rotate continuously.
3-D Hilbert Curve Animation https://www.youtube.com/watch?v=TfJEJidwkBQ
Reference
This module is based on the C code provided in the 2004 article "Programming the Hilbert Curve" by John Skilling,
I was also helped by the discussion in the following stackoverflow post,
which points out a typo in the source code of the paper. The Skilling code
provides two functions TransposetoAxes and AxestoTranspose. In this
case, Transpose refers to a specific packing of the integer that represents
distance along the Hilbert curve (see below for details) and
Axes refer to the n-dimensional coordinates. Below is an excerpt from the
documentation of Skilling's code,
//+++++++++++++++++++++++++++ PUBLIC-DOMAIN SOFTWARE ++++++++++++++++++++++++++ // Functions: TransposetoAxes AxestoTranspose // Purpose: Transform in-place between Hilbert transpose and geometrical axes // Example: b=5 bits for each of n=3 coordinates. // 15-bit Hilbert integer = A B C D E F G H I J K L M N O is stored // as its Transpose // X[0] = A D G J M X[2]| // X[1] = B E H K N <-------> | /X[1] // X[2] = C F I L O axes |/ // high low 0------ X[0] // Axes are stored conveniently as b-bit integers. // Author: John Skilling 20 Apr 2001 to 11 Oct 2003
Change Log
Version 2.0
Version 2.0 introduces some breaking changes.
API Changes
Previous versions transformed a single distance to a vector or a single vector to a distance.
coordinates_from_distance(self, h: int) -> List[int]distance_from_coordinates(self, x_in: List[int]) -> int
In version 2.0 coordinates -> point(s) and we add methods to handle multiple distances or multiple points. The match_type kwarg forces the output type to match the input type and all functions can handle tuples, lists, and ndarrays.
point_from_distance(self, distance: int) -> Iterable[int]points_from_distances(self, distances: Iterable[int], match_type: bool=False) -> Iterable[Iterable[int]]distance_from_point(self, point: Iterable[int]) -> intdistances_from_points(self, points: Iterable[Iterable[int]], match_type: bool=False) -> Iterable[int]
Multiprocessing
The methods that handle multiple distances or multiple points can take advantage of multiple cores. You can control this behavior using the n_procs kwarg when you create an instance of HilbertCurve.