所有可能的矩阵组合

Question

如何获得所有可能的 0 和 1 矩阵？ 例如：

我的 function 定义为

def matrix_combinations(m):

其中 m 是只有零的矩阵

我想要的是所有可能的矩阵，其中 0 和 1 的尺寸与 m 相同

例如： m = [[0, 0, 0], [0, 0, 0]] output 应该是这样的矩阵列表：

[[[0, 0, 0], [0, 0, 0]],
 [[0, 0, 0], [0, 0, 1]],
 [[0, 0, 0], [0, 1, 0]],
 [[0, 0, 0], [0, 1, 1]],
           ...
 [[0, 1, 1], [1, 1, 1]],
 [[1, 1, 1], [1, 1, 1]]]

我不想像 itertools 或 numpy 那样使用 package

Answer 1

我的实现背后的想法是数组的形状无关紧要，您只需要 1 和 0 的所有组合，而要做到这一点，您只需要以二进制计数即可。

我们正在做的是创建与输入矩阵大小相同的二进制数列表，然后更改其形状以匹配输入矩阵。

import numpy as np

def matrix_combinations(m):
    bits = m.shape[0] * m.shape[1]
    amount_numbers = 2**bits
    ret = []
    for i in range(amount_numbers):
        num = [int(x) for x in bin(i)[2:]]
        num = [0] * (bits - len(num)) + num
        ret.append(np.reshape(num, m.shape).tolist())
    return ret

我正在使用 numpy 因为真的很方便改变 arrays 的形状。 如果你不想使用 numpy，假设你的矩阵是二维的，你可以很容易地创建自己的重塑 function。 代码如下所示：

def reshape(list, shape):
    ret = []
    for i in range(shape[0]):
        row = [] 
        for j in range(shape[1]):
            row.append(list[j*i+i])
        ret.append(row)

    return ret

def matrix_combinations(m):
    shape = (len(m),len(m[0]))
    bits = shape[0] * shape[1]
    amount_numbers = 2**bits
    ret = []
    for i in range(0,amount_numbers):
        num = [int(x) for x in bin(i)[2:]]
        num = [0] * (bits - len(num)) + num
        ret.append(reshape(num, shape))
    return ret

Answer 2

您首先需要矩阵的形状 - 实际上最好传递width和height而不是m 。

Numpy 在下面回答，因为它是一个很好的工具，如果你使用矩阵，你绝对应该学习它，这将是值得的。 首先是非numpy的答案。

我将创建0到2**(width*height)范围内的数字。

def matrix_combinations(m):
    height = len(m)
    width = len(m[0])
    size = height * width

    output_matrices = []

    for matrix_values in range(2**(width*height)):
        matrix = []
        for y in range(height):
            row = []
            for x in range(width):
                 last_bit = matrix_values & 1
                 matrix_values >>= 1
                 row.append(last_bit)
            matrix.append(row)
        output_matrices.append(matrix)

    return output_matrices

现在是 numpy 版本。

def matrix_combinations(m):
    indices = np.arange(m.size)
    output_matrices = []
    for value in range(2**m.size):
        matrix_values = np.bitwise_and(value >> indices, 1)
        matrix = matrix_values.reshape(m.shape)
        output_matrices.append(matrix)
    return output_matrices

Answer 3

比矩阵的形状更重要的是元素的数量； 如果它有n = h*w元素，那么你将有2 ** n output 矩阵，只是从 0 到2**n-1的所有数字都写在基数 2 中。下面是一个实现：

def resize(x, height, width):
    return [
        x[i * width : (i+1) * width]
        for i in range(height)
    ]

def all_combinations(m):
    flattened = [x for y in m for x in y]
    n = len(flattened)
    width = len(m[0])
    height = len(m)
    result = []
    form = '{:0' + str(n) + 'b}'
    for i in range(2 ** n):
        s = form.format(i)
        x = [int(c) for c in s]
        result.append(resize(x, height, width))
    return result

Answer 4

这实际上只是m的源数字中的每个二进制掩码，所以我提出了一个非常邪恶的基于字符串的解决方案

def matrix_combinations(m):
    target_length   = str(m).count("0")
    result_template = str(m).replace("0", "{}")
    for numeric in range(1, 2**target_length):
        yield eval(result_template.format(*list(bin(numeric)[2:].zfill(target_length))))

示例使用

>>> _it = matrix_combinations([[0],[0]])
>>> list(_it)
[[[0], [1]], [[1], [0]], [[1], [1]]]
>>> _it = matrix_combinations([[0, 0],[0, 0]])
>>> list(_it)
[[[0, 0], [0, 1]], [[0, 0], [1, 0]], [[0, 0], [1, 1]], [[0, 1], [0, 0]], [[0, 1], [0, 1]], [[0, 1], [1, 0]], [[0, 1], [1, 1]], [[1, 0], [0, 0]], [[1, 0], [0, 1]], [[1, 0], [1, 0]], [[1, 0], [1, 1]], [[1, 1], [0, 0]], [[1, 1], [0, 1]], [[1, 1], [1, 0]], [[1, 1], [1, 1]]]

这通过

• 发现可能有多少排列2**(the count of 0)
• 创建目标矩阵形状的字符串模板

  >>> str([[0, 0],[0, 0]]).replace("0", "{}") '[[{}, {}], [{}, {}]]'

• 用二进制表示每个数字并用前导 0 包装它

  >>> bin(4)[2:].zfill(4) '0100'

• 用值填充模板

  >>> '[[{}, {}], [{}, {}]]'.format(*list(bin(4)[2:].zfill(4))) '[[0, 1], [0, 0]]'

• 评估模板以创建最终结构

---

请注意，如果您尝试将所有这些保留在 memory 中（例如使用列表），您的收集将失败（使用任何解决方案）大约 25 个零（系统 memory 的 5-10G 并且呈指数增长）。但是，如果您处理每个单独的列表（您可以使用像这样的生成器来完成），它几乎可以处理任意数量的零（我的系统在大约 10**7 个零处生成初始列表时开始变得缓慢）

>>> a = next(matrix_combinations([[0]*(10**7)]))
>>>

Answer 5

该解决方案可以采用任何深度和维度的输入矩阵。 通过使用递归生成器 function，作为组合构建过程的一部分，可以更容易地遍历可能的值范围（在本例中为[0, 1] ）：

def prod(d, c = []):
   yield from ([c] if not d else [j for k in d[0] for j in prod(d[1:], c+[k])])

def matrix_combinations(m):
   for i in prod([[0, 1] if not i else [*matrix_combinations(i)] for i in m]):
      yield list(i)

print(list(matrix_combinations([[0, 0, 0], [0, 0, 0]])))
print(list(matrix_combinations([[0, [0, [0, 0]], 0], [0, 0, [0]]])))

Output：

[[[0, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 1]], [[0, 0, 0], [0, 1, 0]], [[0, 0, 0], [0, 1, 1]], [[0, 0, 0], [1, 0, 0]], [[0, 0, 0], [1, 0, 1]], [[0, 0, 0], [1, 1, 0]], [[0, 0, 0], [1, 1, 1]], [[0, 0, 1], [0, 0, 0]], [[0, 0, 1], [0, 0, 1]], [[0, 0, 1], [0, 1, 0]], [[0, 0, 1], [0, 1, 1]], [[0, 0, 1], [1, 0, 0]], [[0, 0, 1], [1, 0, 1]], [[0, 0, 1], [1, 1, 0]], [[0, 0, 1], [1, 1, 1]], [[0, 1, 0], [0, 0, 0]], [[0, 1, 0], [0, 0, 1]], [[0, 1, 0], [0, 1, 0]], [[0, 1, 0], [0, 1, 1]], [[0, 1, 0], [1, 0, 0]], [[0, 1, 0], [1, 0, 1]], [[0, 1, 0], [1, 1, 0]], [[0, 1, 0], [1, 1, 1]], [[0, 1, 1], [0, 0, 0]], [[0, 1, 1], [0, 0, 1]], [[0, 1, 1], [0, 1, 0]], [[0, 1, 1], [0, 1, 1]], [[0, 1, 1], [1, 0, 0]], [[0, 1, 1], [1, 0, 1]], [[0, 1, 1], [1, 1, 0]], [[0, 1, 1], [1, 1, 1]], [[1, 0, 0], [0, 0, 0]], [[1, 0, 0], [0, 0, 1]], [[1, 0, 0], [0, 1, 0]], [[1, 0, 0], [0, 1, 1]], [[1, 0, 0], [1, 0, 0]], [[1, 0, 0], [1, 0, 1]], [[1, 0, 0], [1, 1, 0]], [[1, 0, 0], [1, 1, 1]], [[1, 0, 1], [0, 0, 0]], [[1, 0, 1], [0, 0, 1]], [[1, 0, 1], [0, 1, 0]], [[1, 0, 1], [0, 1, 1]], [[1, 0, 1], [1, 0, 0]], [[1, 0, 1], [1, 0, 1]], [[1, 0, 1], [1, 1, 0]], [[1, 0, 1], [1, 1, 1]], [[1, 1, 0], [0, 0, 0]], [[1, 1, 0], [0, 0, 1]], [[1, 1, 0], [0, 1, 0]], [[1, 1, 0], [0, 1, 1]], [[1, 1, 0], [1, 0, 0]], [[1, 1, 0], [1, 0, 1]], [[1, 1, 0], [1, 1, 0]], [[1, 1, 0], [1, 1, 1]], [[1, 1, 1], [0, 0, 0]], [[1, 1, 1], [0, 0, 1]], [[1, 1, 1], [0, 1, 0]], [[1, 1, 1], [0, 1, 1]], [[1, 1, 1], [1, 0, 0]], [[1, 1, 1], [1, 0, 1]], [[1, 1, 1], [1, 1, 0]], [[1, 1, 1], [1, 1, 1]]]
[[[0, [0, [0, 0]], 0], [0, 0, [0]]], [[0, [0, [0, 0]], 0], [0, 0, [1]]], [[0, [0, [0, 0]], 0], [0, 1, [0]]], [[0, [0, [0, 0]], 0], [0, 1, [1]]], [[0, [0, [0, 0]], 0], [1, 0, [0]]], [[0, [0, [0, 0]], 0], [1, 0, [1]]], [[0, [0, [0, 0]], 0], [1, 1, [0]]], [[0, [0, [0, 0]], 0], [1, 1, [1]]], [[0, [0, [0, 0]], 1], [0, 0, [0]]], [[0, [0, [0, 0]], 1], [0, 0, [1]]], [[0, [0, [0, 0]], 1], [0, 1, [0]]], [[0, [0, [0, 0]], 1], [0, 1, [1]]], [[0, [0, [0, 0]], 1], [1, 0, [0]]], [[0, [0, [0, 0]], 1], [1, 0, [1]]], [[0, [0, [0, 0]], 1], [1, 1, [0]]], [[0, [0, [0, 0]], 1], [1, 1, [1]]], [[0, [0, [0, 1]], 0], [0, 0, [0]]], [[0, [0, [0, 1]], 0], [0, 0, [1]]], [[0, [0, [0, 1]], 0], [0, 1, [0]]], [[0, [0, [0, 1]], 0], [0, 1, [1]]], [[0, [0, [0, 1]], 0], [1, 0, [0]]], [[0, [0, [0, 1]], 0], [1, 0, [1]]], [[0, [0, [0, 1]], 0], [1, 1, [0]]], [[0, [0, [0, 1]], 0], [1, 1, [1]]], [[0, [0, [0, 1]], 1], [0, 0, [0]]], [[0, [0, [0, 1]], 1], [0, 0, [1]]], [[0, [0, [0, 1]], 1], [0, 1, [0]]], [[0, [0, [0, 1]], 1], [0, 1, [1]]], [[0, [0, [0, 1]], 1], [1, 0, [0]]], [[0, [0, [0, 1]], 1], [1, 0, [1]]], [[0, [0, [0, 1]], 1], [1, 1, [0]]], [[0, [0, [0, 1]], 1], [1, 1, [1]]], [[0, [0, [1, 0]], 0], [0, 0, [0]]], [[0, [0, [1, 0]], 0], [0, 0, [1]]], [[0, [0, [1, 0]], 0], [0, 1, [0]]], [[0, [0, [1, 0]], 0], [0, 1, [1]]], [[0, [0, [1, 0]], 0], [1, 0, [0]]], [[0, [0, [1, 0]], 0], [1, 0, [1]]], [[0, [0, [1, 0]], 0], [1, 1, [0]]], [[0, [0, [1, 0]], 0], [1, 1, [1]]], [[0, [0, [1, 0]], 1], [0, 0, [0]]], [[0, [0, [1, 0]], 1], [0, 0, [1]]], [[0, [0, [1, 0]], 1], [0, 1, [0]]], [[0, [0, [1, 0]], 1], [0, 1, [1]]], [[0, [0, [1, 0]], 1], [1, 0, [0]]], [[0, [0, [1, 0]], 1], [1, 0, [1]]], [[0, [0, [1, 0]], 1], [1, 1, [0]]], [[0, [0, [1, 0]], 1], [1, 1, [1]]], [[0, [0, [1, 1]], 0], [0, 0, [0]]], [[0, [0, [1, 1]], 0], [0, 0, [1]]], [[0, [0, [1, 1]], 0], [0, 1, [0]]], [[0, [0, [1, 1]], 0], [0, 1, [1]]], [[0, [0, [1, 1]], 0], [1, 0, [0]]], [[0, [0, [1, 1]], 0], [1, 0, [1]]], [[0, [0, [1, 1]], 0], [1, 1, [0]]], [[0, [0, [1, 1]], 0], [1, 1, [1]]], [[0, [0, [1, 1]], 1], [0, 0, [0]]], [[0, [0, [1, 1]], 1], [0, 0, [1]]], [[0, [0, [1, 1]], 1], [0, 1, [0]]], [[0, [0, [1, 1]], 1], [0, 1, [1]]], [[0, [0, [1, 1]], 1], [1, 0, [0]]], [[0, [0, [1, 1]], 1], [1, 0, [1]]], [[0, [0, [1, 1]], 1], [1, 1, [0]]], [[0, [0, [1, 1]], 1], [1, 1, [1]]], [[0, [1, [0, 0]], 0], [0, 0, [0]]], [[0, [1, [0, 0]], 0], [0, 0, [1]]], [[0, [1, [0, 0]], 0], [0, 1, [0]]], [[0, [1, [0, 0]], 0], [0, 1, [1]]], [[0, [1, [0, 0]], 0], [1, 0, [0]]], [[0, [1, [0, 0]], 0], [1, 0, [1]]], [[0, [1, [0, 0]], 0], [1, 1, [0]]], [[0, [1, [0, 0]], 0], [1, 1, [1]]], [[0, [1, [0, 0]], 1], [0, 0, [0]]], [[0, [1, [0, 0]], 1], [0, 0, [1]]], [[0, [1, [0, 0]], 1], [0, 1, [0]]], [[0, [1, [0, 0]], 1], [0, 1, [1]]], [[0, [1, [0, 0]], 1], [1, 0, [0]]], [[0, [1, [0, 0]], 1], [1, 0, [1]]], [[0, [1, [0, 0]], 1], [1, 1, [0]]], [[0, [1, [0, 0]], 1], [1, 1, [1]]], [[0, [1, [0, 1]], 0], [0, 0, [0]]], [[0, [1, [0, 1]], 0], [0, 0, [1]]], [[0, [1, [0, 1]], 0], [0, 1, [0]]], [[0, [1, [0, 1]], 0], [0, 1, [1]]], [[0, [1, [0, 1]], 0], [1, 0, [0]]], [[0, [1, [0, 1]], 0], [1, 0, [1]]], [[0, [1, [0, 1]], 0], [1, 1, [0]]], [[0, [1, [0, 1]], 0], [1, 1, [1]]], [[0, [1, [0, 1]], 1], [0, 0, [0]]], [[0, [1, [0, 1]], 1], [0, 0, [1]]], [[0, [1, [0, 1]], 1], [0, 1, [0]]], [[0, [1, [0, 1]], 1], [0, 1, [1]]], [[0, [1, [0, 1]], 1], [1, 0, [0]]], [[0, [1, [0, 1]], 1], [1, 0, [1]]], [[0, [1, [0, 1]], 1], [1, 1, [0]]], [[0, [1, [0, 1]], 1], [1, 1, [1]]], [[0, [1, [1, 0]], 0], [0, 0, [0]]], [[0, [1, [1, 0]], 0], [0, 0, [1]]], [[0, [1, [1, 0]], 0], [0, 1, [0]]], [[0, [1, [1, 0]], 0], [0, 1, [1]]], [[0, [1, [1, 0]], 0], [1, 0, [0]]], [[0, [1, [1, 0]], 0], [1, 0, [1]]], [[0, [1, [1, 0]], 0], [1, 1, [0]]], [[0, [1, [1, 0]], 0], [1, 1, [1]]], [[0, [1, [1, 0]], 1], [0, 0, [0]]], [[0, [1, [1, 0]], 1], [0, 0, [1]]], [[0, [1, [1, 0]], 1], [0, 1, [0]]], [[0, [1, [1, 0]], 1], [0, 1, [1]]], [[0, [1, [1, 0]], 1], [1, 0, [0]]], [[0, [1, [1, 0]], 1], [1, 0, [1]]], [[0, [1, [1, 0]], 1], [1, 1, [0]]], [[0, [1, [1, 0]], 1], [1, 1, [1]]], [[0, [1, [1, 1]], 0], [0, 0, [0]]], [[0, [1, [1, 1]], 0], [0, 0, [1]]], [[0, [1, [1, 1]], 0], [0, 1, [0]]], [[0, [1, [1, 1]], 0], [0, 1, [1]]], [[0, [1, [1, 1]], 0], [1, 0, [0]]], [[0, [1, [1, 1]], 0], [1, 0, [1]]], [[0, [1, [1, 1]], 0], [1, 1, [0]]], [[0, [1, [1, 1]], 0], [1, 1, [1]]], [[0, [1, [1, 1]], 1], [0, 0, [0]]], [[0, [1, [1, 1]], 1], [0, 0, [1]]], [[0, [1, [1, 1]], 1], [0, 1, [0]]], [[0, [1, [1, 1]], 1], [0, 1, [1]]], [[0, [1, [1, 1]], 1], [1, 0, [0]]], [[0, [1, [1, 1]], 1], [1, 0, [1]]], [[0, [1, [1, 1]], 1], [1, 1, [0]]], [[0, [1, [1, 1]], 1], [1, 1, [1]]], [[1, [0, [0, 0]], 0], [0, 0, [0]]], [[1, [0, [0, 0]], 0], [0, 0, [1]]], [[1, [0, [0, 0]], 0], [0, 1, [0]]], [[1, [0, [0, 0]], 0], [0, 1, [1]]], [[1, [0, [0, 0]], 0], [1, 0, [0]]], [[1, [0, [0, 0]], 0], [1, 0, [1]]], [[1, [0, [0, 0]], 0], [1, 1, [0]]], [[1, [0, [0, 0]], 0], [1, 1, [1]]], [[1, [0, [0, 0]], 1], [0, 0, [0]]], [[1, [0, [0, 0]], 1], [0, 0, [1]]], [[1, [0, [0, 0]], 1], [0, 1, 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