hilbertcurve 2.0.5
Construct Hilbert Curves.
License: MIT
Requires Python: >=3.6
Keywords: fractal space-filling-curves hilbert hilbert-curve python python3

.. image:: https://travis-ci.com/galtay/hilbertcurve.svg?branch=develop
    :target: https://travis-ci.com/galtay/hilbertcurve

.. contents:: Table of Contents
   :depth: 2


============
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,

.. code-block:: bash

  pip install hilbertcurve

You can calculate points given distances along a hilbert curve,

.. code-block:: python

  >>> 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,

.. code-block:: python

  >>> 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]) = 3

=========================
Inputs 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]) -> int``
* ``distances_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,

.. code-block:: python

  >>> 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.ndarray
  
=========================
Multiprocessing
=========================

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``. 

.. code-block:: bash

  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 use

A 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,

.. code-block:: python

  >>> 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!

.. code-block:: python

  >>> 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
=======


.. figure:: https://raw.githubusercontent.com/galtay/hilbertcurve/main/n2_p3.png

   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.

.. figure:: https://img.youtube.com/vi/TfJEJidwkBQ/0.jpg

   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,

* http://adsabs.harvard.edu/abs/2004AIPC..707..381S

I was also helped by the discussion in the following stackoverflow post,

* `mapping-n-dimensional-value-to-a-point-on-hilbert-curve`_

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]) -> int``
* ``distances_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`.



.. _Hilbert curve: https://en.wikipedia.org/wiki/Hilbert_curve
.. _hypercube: https://en.wikipedia.org/wiki/Hypercube
.. _arbitrarily large integers in Python: https://docs.python.org/3.3/library/stdtypes.html#numeric-types-int-float-complex
.. _an estimate of the number of atoms in the observable universe: https://www.universetoday.com/36302/atoms-in-the-universe
.. _mapping-n-dimensional-value-to-a-point-on-hilbert-curve: http://stackoverflow.com/questions/499166/mapping-n-dimensional-value-to-a-point-on-hilbert-curve