# Copyright 2022-2023 XProbe Inc.
# derived from copyright 1999-2021 Alibaba Group Holding Ltd.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
import functools
import itertools
import operator
import numpy as np
from ..datasource.array import array
from ..datasource.empty import empty
# Internal functions to eliminate the overhead of repeated dispatch in one of
# the two possible paths inside mt.block.
# Use getattr to protect against __array_function__ being disabled.
_size = getattr(np.size, "__wrapped__", np.size)
_ndim = getattr(np.ndim, "__wrapped__", np.ndim)
def _block_format_index(index):
"""
Convert a list of indices ``[0, 1, 2]`` into ``"arrays[0][1][2]"``.
"""
idx_str = "".join("[{}]".format(i) for i in index if i is not None)
return "arrays" + idx_str
def _block_check_depths_match(arrays, parent_index=[]):
"""
Recursive function checking that the depths of nested lists in `arrays`
all match. Mismatch raises a ValueError as described in the block
docstring below.
The entire index (rather than just the depth) needs to be calculated
for each innermost list, in case an error needs to be raised, so that
the index of the offending list can be printed as part of the error.
Parameters
----------
arrays : nested list of arrays
The arrays to check
parent_index : list of int
The full index of `arrays` within the nested lists passed to
`_block_check_depths_match` at the top of the recursion.
Returns
-------
first_index : list of int
The full index of an element from the bottom of the nesting in
`arrays`. If any element at the bottom is an empty list, this will
refer to it, and the last index along the empty axis will be None.
max_arr_ndim : int
The maximum of the ndims of the arrays nested in `arrays`.
final_size: int
The number of elements in the final array. This is used the motivate
the choice of algorithm used using benchmarking wisdom.
"""
if type(arrays) is tuple:
# not strictly necessary, but saves us from:
# - more than one way to do things - no point treating tuples like
# lists
# - horribly confusing behaviour that results when tuples are
# treated like ndarray
raise TypeError(
"{} is a tuple. "
"Only lists can be used to arrange blocks, and mt.block does "
"not allow implicit conversion from tuple to ndarray.".format(
_block_format_index(parent_index)
)
)
elif type(arrays) is list and len(arrays) > 0:
idxs_ndims = (
_block_check_depths_match(arr, parent_index + [i])
for i, arr in enumerate(arrays)
)
first_index, max_arr_ndim, final_size = next(idxs_ndims)
for index, ndim, size in idxs_ndims:
final_size += size
if ndim > max_arr_ndim:
max_arr_ndim = ndim
if len(index) != len(first_index):
raise ValueError(
"List depths are mismatched. First element was at depth "
"{}, but there is an element at depth {} ({})".format(
len(first_index), len(index), _block_format_index(index)
)
)
# propagate our flag that indicates an empty list at the bottom
if index[-1] is None:
first_index = index
return first_index, max_arr_ndim, final_size
elif type(arrays) is list and len(arrays) == 0:
# We've 'bottomed out' on an empty list
return parent_index + [None], 0, 0
else:
# We've 'bottomed out' - arrays is either a scalar or an array
size = _size(arrays)
return parent_index, _ndim(arrays), size
def _atleast_nd(a, ndim):
# Ensures `a` has at least `ndim` dimensions by prepending
# ones to `a.shape` as necessary
return array(a, ndmin=ndim, copy=False)
def _accumulate(values):
return list(itertools.accumulate(values))
def _concatenate_shapes(shapes, axis):
"""Given array shapes, return the resulting shape and slices prefixes.
These help in nested concatenation.
Returns
-------
shape: tuple of int
This tuple satisfies:
```
shape, _ = _concatenate_shapes([arr.shape for shape in arrs], axis)
shape == concatenate(arrs, axis).shape
```
slice_prefixes: tuple of (slice(start, end), )
For a list of arrays being concatenated, this returns the slice
in the larger array at axis that needs to be sliced into.
For example, the following holds:
```
ret = concatenate([a, b, c], axis)
_, (sl_a, sl_b, sl_c) = concatenate_slices([a, b, c], axis)
ret[(slice(None),) * axis + sl_a] == a
ret[(slice(None),) * axis + sl_b] == b
ret[(slice(None),) * axis + sl_c] == c
```
These are called slice prefixes since they are used in the recursive
blocking algorithm to compute the left-most slices during the
recursion. Therefore, they must be prepended to rest of the slice
that was computed deeper in the recursion.
These are returned as tuples to ensure that they can quickly be added
to existing slice tuple without creating a new tuple every time.
"""
# Cache a result that will be reused.
shape_at_axis = [shape[axis] for shape in shapes]
# Take a shape, any shape
first_shape = shapes[0]
first_shape_pre = first_shape[:axis]
first_shape_post = first_shape[axis + 1 :]
if any(
shape[:axis] != first_shape_pre or shape[axis + 1 :] != first_shape_post
for shape in shapes
):
raise ValueError("Mismatched array shapes in block along axis {}.".format(axis))
shape = first_shape_pre + (sum(shape_at_axis),) + first_shape[axis + 1 :]
offsets_at_axis = _accumulate(shape_at_axis)
slice_prefixes = [
(slice(start, end),)
for start, end in zip([0] + offsets_at_axis, offsets_at_axis)
]
return shape, slice_prefixes
def _block_info_recursion(arrays, max_depth, result_ndim, depth=0):
"""
Returns the shape of the final array, along with a list
of slices and a list of arrays that can be used for assignment inside the
new array
Parameters
----------
arrays : nested list of arrays
The arrays to check
max_depth : list of int
The number of nested lists
result_ndim: int
The number of dimensions in thefinal array.
Returns
-------
shape : tuple of int
The shape that the final array will take on.
slices: list of tuple of slices
The slices into the full array required for assignment. These are
required to be prepended with ``(Ellipsis, )`` to obtain to correct
final index.
arrays: list of ndarray
The data to assign to each slice of the full array
"""
if depth < max_depth:
shapes, slices, arrays = zip(
*[
_block_info_recursion(arr, max_depth, result_ndim, depth + 1)
for arr in arrays
]
)
axis = result_ndim - max_depth + depth
shape, slice_prefixes = _concatenate_shapes(shapes, axis)
# Prepend the slice prefix and flatten the slices
slices = [
slice_prefix + the_slice
for slice_prefix, inner_slices in zip(slice_prefixes, slices)
for the_slice in inner_slices
]
# Flatten the array list
arrays = functools.reduce(operator.add, arrays)
return shape, slices, arrays
else:
# We've 'bottomed out' - arrays is either a scalar or an array
# type(arrays) is not list
# Return the slice and the array inside a list to be consistent with
# the recursive case.
arr = _atleast_nd(arrays, result_ndim)
return arr.shape, [()], [arr]
def _block(arrays, max_depth, result_ndim, depth=0):
"""
Internal implementation of block based on repeated concatenation.
`arrays` is the argument passed to
block. `max_depth` is the depth of nested lists within `arrays` and
`result_ndim` is the greatest of the dimensions of the arrays in
`arrays` and the depth of the lists in `arrays` (see block docstring
for details).
"""
from ..merge.concatenate import concatenate
if depth < max_depth:
arrs = [_block(arr, max_depth, result_ndim, depth + 1) for arr in arrays]
return concatenate(arrs, axis=-(max_depth - depth))
else:
# We've 'bottomed out' - arrays is either a scalar or an array
# type(arrays) is not list
return _atleast_nd(arrays, result_ndim)
[docs]def block(arrays):
"""
Assemble an nd-array from nested lists of blocks.
Blocks in the innermost lists are concatenated (see `concatenate`) along
the last dimension (-1), then these are concatenated along the
second-last dimension (-2), and so on until the outermost list is reached.
Blocks can be of any dimension, but will not be broadcasted using the normal
rules. Instead, leading axes of size 1 are inserted, to make ``block.ndim``
the same for all blocks. This is primarily useful for working with scalars,
and means that code like ``mt.block([v, 1])`` is valid, where
``v.ndim == 1``.
When the nested list is two levels deep, this allows block matrices to be
constructed from their components.
.. versionadded:: 1.13.0
Parameters
----------
arrays : nested list of array_like or scalars (but not tuples)
If passed a single ndarray or scalar (a nested list of depth 0), this
is returned unmodified (and not copied).
Elements shapes must match along the appropriate axes (without
broadcasting), but leading 1s will be prepended to the shape as
necessary to make the dimensions match.
Returns
-------
block_array : Tensor
The array assembled from the given blocks.
The dimensionality of the output is equal to the greatest of:
* the dimensionality of all the inputs
* the depth to which the input list is nested
Raises
------
ValueError
* If list depths are mismatched - for instance, ``[[a, b], c]`` is
illegal, and should be spelt ``[[a, b], [c]]``
* If lists are empty - for instance, ``[[a, b], []]``
See Also
--------
concatenate : Join a sequence of arrays along an existing axis.
stack : Join a sequence of arrays along a new axis.
vstack : Stack arrays in sequence vertically (row wise).
hstack : Stack arrays in sequence horizontally (column wise).
dstack : Stack arrays in sequence depth wise (along third axis).
column_stack : Stack 1-D arrays as columns into a 2-D array.
vsplit : Split an array into multiple sub-arrays vertically (row-wise).
Notes
-----
When called with only scalars, ``mt.block`` is equivalent to an ndarray
call. So ``mt.block([[1, 2], [3, 4]])`` is equivalent to
``mt.array([[1, 2], [3, 4]])``.
This function does not enforce that the blocks lie on a fixed grid.
``mt.block([[a, b], [c, d]])`` is not restricted to arrays of the form::
AAAbb
AAAbb
cccDD
But is also allowed to produce, for some ``a, b, c, d``::
AAAbb
AAAbb
cDDDD
Since concatenation happens along the last axis first, `block` is _not_
capable of producing the following directly::
AAAbb
cccbb
cccDD
Matlab's "square bracket stacking", ``[A, B, ...; p, q, ...]``, is
equivalent to ``mt.block([[A, B, ...], [p, q, ...]])``.
Examples
--------
The most common use of this function is to build a block matrix
>>> import mars.tensor as mt
>>> A = mt.eye(2) * 2
>>> B = mt.eye(3) * 3
>>> mt.block([
... [A, mt.zeros((2, 3))],
... [mt.ones((3, 2)), B ]
... ]).execute()
array([[2., 0., 0., 0., 0.],
[0., 2., 0., 0., 0.],
[1., 1., 3., 0., 0.],
[1., 1., 0., 3., 0.],
[1., 1., 0., 0., 3.]])
With a list of depth 1, `block` can be used as `hstack`
>>> mt.block([1, 2, 3]).execute() # hstack([1, 2, 3])
array([1, 2, 3])
>>> a = mt.array([1, 2, 3])
>>> b = mt.array([2, 3, 4])
>>> mt.block([a, b, 10]).execute() # hstack([a, b, 10])
array([ 1, 2, 3, 2, 3, 4, 10])
>>> A = mt.ones((2, 2), int)
>>> B = 2 * A
>>> mt.block([A, B]).execute() # hstack([A, B])
array([[1, 1, 2, 2],
[1, 1, 2, 2]])
With a list of depth 2, `block` can be used in place of `vstack`:
>>> a = mt.array([1, 2, 3])
>>> b = mt.array([2, 3, 4])
>>> mt.block([[a], [b]]).execute() # vstack([a, b])
array([[1, 2, 3],
[2, 3, 4]])
>>> A = mt.ones((2, 2), int)
>>> B = 2 * A
>>> mt.block([[A], [B]]).execute() # vstack([A, B])
array([[1, 1],
[1, 1],
[2, 2],
[2, 2]])
It can also be used in places of `atleast_1d` and `atleast_2d`
>>> a = mt.array(0)
>>> b = mt.array([1])
>>> mt.block([a]).execute() # atleast_1d(a)
array([0])
>>> mt.block([b]).execute() # atleast_1d(b)
array([1])
>>> mt.block([[a]]).execute() # atleast_2d(a)
array([[0]])
>>> mt.block([[b]]).execute() # atleast_2d(b)
array([[1]])
"""
arrays, list_ndim, result_ndim, final_size = _block_setup(arrays)
# It was found through benchmarking that making an array of final size
# around 256x256 was faster by straight concatenation on a
# i7-7700HQ processor and dual channel ram 2400MHz.
# It didn't seem to matter heavily on the dtype used.
#
# A 2D array using repeated concatenation requires 2 copies of the array.
#
# The fastest algorithm will depend on the ratio of CPU power to memory
# speed.
# One can monitor the results of the benchmark
# https://pv.github.io/numpy-bench/#bench_shape_base.Block2D.time_block2d
# to tune this parameter until a C version of the `_block_info_recursion`
# algorithm is implemented which would likely be faster than the python
# version.
if list_ndim * final_size > (2 * 512 * 512):
return _block_slicing(arrays, list_ndim, result_ndim)
else:
return _block_concatenate(arrays, list_ndim, result_ndim)
# These helper functions are mostly used for testing.
# They allow us to write tests that directly call `_block_slicing`
# or `_block_concatenate` without blocking large arrays to force the wisdom
# to trigger the desired path.
def _block_setup(arrays):
"""
Returns
(`arrays`, list_ndim, result_ndim, final_size)
"""
bottom_index, arr_ndim, final_size = _block_check_depths_match(arrays)
list_ndim = len(bottom_index)
if bottom_index and bottom_index[-1] is None:
raise ValueError(
"List at {} cannot be empty".format(_block_format_index(bottom_index))
)
result_ndim = max(arr_ndim, list_ndim)
return arrays, list_ndim, result_ndim, final_size
def _block_slicing(arrays, list_ndim, result_ndim):
shape, slices, arrays = _block_info_recursion(arrays, list_ndim, result_ndim)
dtype = np.result_type(*[arr.dtype for arr in arrays])
# Test preferring F only in the case that all input arrays are F
F_order = all(arr.flags["F_CONTIGUOUS"] for arr in arrays)
C_order = all(arr.flags["C_CONTIGUOUS"] for arr in arrays)
order = "F" if F_order and not C_order else "C"
result = empty(shape=shape, dtype=dtype, order=order)
# Note: In a c implementation, the function
# PyArray_CreateMultiSortedStridePerm could be used for more advanced
# guessing of the desired order.
for the_slice, arr in zip(slices, arrays):
result[(Ellipsis,) + the_slice] = arr
return result
def _block_concatenate(arrays, list_ndim, result_ndim):
result = _block(arrays, list_ndim, result_ndim)
if list_ndim == 0:
# Catch an edge case where _block returns a view because
# `arrays` is a single mars array and not a list of mars arrays.
# This might copy scalars or lists twice, but this isn't a likely
# usecase for those interested in performance
result = result.copy()
return result