生成 n 个项目的 k 元组的算法，使得每个项目至少使用一次 (k>n)

Question

给定一组 n 项，我想生成我们可以从该组中选择的所有方式（带替换） k 次，这样顺序很重要，并且每个元素至少使用一次。所以 k>=n 有任何有效的安排。如果 k=n，这只是一个排列。所以 k>n 有点像排列的扩展，但我不知道它叫什么。

当然很容易获得一种算法，尽管它的速度非常慢：只需遍历所有可能的选择，并将没有每个元素的选择至少扔一次。因此，要使事情变得高效，就需要类似于遍历排列的技巧，或者将其分解为子问题，我们可以直接使用现有的排列算法。

我试图通过使用 python 执行以下操作来将其分解为一个排列和组合问题。

import itertools

def func(inputSet,k):
    n = len(inputSet)
    assert(k>n)
    # first, guarantee we have each element once
    for p in itertools.permutations(inputSet):
        # now select (k-n) locations to insert other elements
        for c in itertools.combinations_with_replacement(range(n+1),k-n):
            insertions = [c[i]+i for i in range(len(c))]
            out = list(p)
            for index in insertions:
                out.insert(index,0)
            # now select values to put in those locations
            for vals in itertools.product(inputSet,repeat=len(insertions)):
                for i in xrange(len(insertions)):
                    out[c[i]] = vals[i]
                yield tuple(out)

但不同的插入可能会产生相同的结果，因此第一次插入可能不会从正确的路径开始。我可以添加条件来检查这些情况并过滤掉一些结果，但是用于组合迭代问题的算法可能不是最有效的算法。

这个“置换扩展”有名字吗？
什么是迭代排列的有效算法？

Answer 1

就其名称而言，这与排列相同，只是框架略有不同。考虑元素 P 的集合，您实际上是在要求生成集合 P 中与 P 的 (k-n) 个元素联合的所有排列，可以通过 itertools.combinations_with_replacement 找到。

要生成实际的排列，您可以使用list(set(itertools.permutations)) 或more_itertools.distinct_permutations：https://more-itertools.readthedocs.io/en/latest/api.html#more_itertools.distinct_permutations

将其放入实际代码中

>>> x = [1,2,3]
>>> k = 5
>>> results = set()
>>> for y in itertools.combinations_with_replacement(x, k - len(x)):
...   for z in itertools.permutations(x + list(y)):
...     results.add(z)
...
>>> results
set([(1, 1, 1, 2, 3), (1, 3, 3, 1, 2), (1, 2, 3, 3, 2), (3, 3, 2, 1, 3), (1, 3, 3, 2, 2), (1, 1, 2, 2, 3), (3, 1, 2, 3, 2), (1, 1, 3, 2, 3), (1, 3, 1, 2, 2), (1, 2, 2, 1, 3), (3, 2, 1, 2, 2), (3, 1, 2, 1, 2), (3, 2, 1, 3, 1), (3, 1, 1, 3, 2), (2, 3, 3, 2, 1), (1, 2, 1, 3, 3), (3, 1, 2, 3, 3), (2, 3, 2, 1, 1), (2, 3, 2, 2, 1), (1, 2, 2, 3, 3), (2, 1, 3, 2, 3), (2, 2, 2, 3, 1), (1, 3, 2, 2, 3), (2, 3, 3, 1, 1), (1, 2, 1, 1, 3), (3, 2, 2, 1, 1), (2, 1, 2, 2, 3), (2, 2, 1, 3, 1), (1, 3, 3, 2, 3), (2, 3, 3, 1, 3), (3, 2, 3, 2, 1), (3, 2, 3, 1, 1), (2, 1, 1, 2, 3), (3, 3, 1, 1, 2), (3, 2, 2, 2, 1), (2, 2, 3, 2, 1), (1, 3, 1, 2, 3), (2, 2, 3, 1, 2), (3, 1, 2, 1, 3), (2, 1, 1, 3, 3), (3, 3, 2, 2, 1), (1, 1, 2, 3, 1), (1, 2, 2, 3, 2), (3, 2, 1, 1, 2), (2, 1, 3, 2, 2), (1, 3, 2, 2, 2), (3, 1, 1, 1, 2), (3, 3, 2, 1, 1), (2, 3, 3, 1, 2), (3, 2, 1, 3, 2), (1, 2, 1, 3, 1), (2, 3, 1, 2, 3), (1, 2, 3, 2, 1), (3, 1, 3, 1, 2), (3, 3, 1, 2, 2), (1, 2, 3, 1, 1), (2, 2, 1, 1, 3), (2, 1, 1, 3, 2), (1, 1, 2, 3, 2), (3, 2, 1, 1, 3), (2, 1, 3, 2, 1), (2, 1, 1, 3, 1), (1, 3, 2, 2, 1), (1, 3, 2, 1, 1), (3, 2, 2, 1, 3), (2, 2, 3, 3, 1), (3, 1, 1, 2, 1), (2, 2, 1, 3, 3), (1, 3, 3, 2, 1), (3, 2, 3, 1, 2), (3, 1, 2, 3, 1), (2, 2, 2, 1, 3), (1, 1, 3, 1, 2), (1, 1, 2, 1, 3), (2, 1, 3, 3, 2), (3, 3, 1, 2, 3), (1, 3, 2, 3, 2), (3, 3, 2, 3, 1), (3, 1, 3, 2, 2), (2, 1, 3, 1, 1), (1, 1, 2, 3, 3), (2, 1, 3, 1, 2), (1, 3, 2, 1, 2), (1, 2, 1, 2, 3), (3, 2, 2, 1, 2), (3, 1, 1, 2, 2), (2, 2, 1, 3, 2), (2, 3, 2, 3, 1), (1, 1, 1, 3, 2), (2, 3, 1, 2, 1), (1, 2, 3, 1, 3), (2, 3, 1, 1, 1), (1, 2, 3, 2, 3), (2, 1, 2, 3, 3), (3, 2, 1, 2, 3), (1, 2, 2, 2, 3), (3, 2, 1, 2, 1), (2, 1, 1, 1, 3), (1, 3, 2, 3, 3), (1, 1, 3, 3, 2), (3, 2, 2, 3, 1), (3, 1, 3, 2, 3), (2, 1, 2, 1, 3), (1, 3, 3, 3, 2), (3, 2, 3, 3, 1), (2, 2, 3, 1, 3), (3, 2, 1, 1, 1), (2, 1, 3, 1, 3), (1, 2, 1, 3, 2), (3, 3, 1, 3, 2), (1, 3, 2, 1, 3), (2, 3, 1, 3, 2), (3, 1, 1, 2, 3), (2, 3, 3, 3, 1), (1, 1, 3, 2, 1), (2, 3, 1, 1, 2), (1, 2, 3, 2, 2), (2, 1, 2, 3, 2), (1, 3, 1, 1, 2), (3, 1, 3, 3, 2), (3, 1, 2, 2, 2), (3, 3, 1, 2, 1), (3, 3, 3, 1, 2), (3, 2, 3, 1, 3), (1, 3, 1, 3, 2), (2, 3, 2, 1, 3), (2, 1, 3, 3, 3), (1, 2, 2, 3, 1), (2, 3, 1, 3, 3), (1, 2, 3, 3, 1), (2, 3, 1, 2, 2), (3, 3, 2, 1, 2), (2, 3, 1, 3, 1), (3, 2, 1, 3, 3), (1, 1, 3, 2, 2), (2, 3, 1, 1, 3), (2, 1, 2, 3, 1), (1, 2, 3, 3, 3), (1, 3, 1, 2, 1), (3, 1, 2, 2, 3), (3, 1, 2, 2, 1), (2, 1, 3, 3, 1), (3, 1, 2, 1, 1), (1, 3, 2, 3, 1), (1, 2, 3, 1, 2), (3, 1, 3, 2, 1), (2, 2, 1, 2, 3), (2, 3, 2, 1, 2), (3, 3, 3, 2, 1), (2, 2, 3, 1, 1)])

请注意，这组合起来很快就会爆炸，但是因为itertools.combinations_with_replacement 和itertools.permutations 都返回生成器，所以您也可以yield 结果。您也可以自己递归编写，但我个人认为这不太令人满意。

我相信在这里使用distinct_permutations 也足够了，您最终会得到一个完全不同的结果列表，因为外部循环的每次迭代都会导致元素的频率签名不同。

Answer 2

这是一个额外的解决方案，它也按字典顺序迭代。

import itertools

def _recurse_extended_perm(remaining,inputList,requireList):
    if remaining==len(requireList):
        for p in itertools.permutations(requireList):
            yield p
    elif remaining==1:
        for x in inputList:
            yield [x]
    else:
        for x in inputList:
            req = list(requireList)
            if x in req:
                req.remove(x)
            for seq in _recurse_extended_perm(remaining-1,inputList,req):
                out = [x]
                out.extend(seq)
                yield out

def extended_permutation(seq,k):
    inputList = list(seq)
    inputList.sort()
    assert(k>len(inputList))
    for seq in _recurse_extended_perm(k,inputList,inputList):
        yield tuple(seq)

Answer 3

我们可以递归地生成列表。这是一个无模块的实现，它也给了我们一点灵活性：

def multisets_with_at_least_m(inputList, k, m):
  def f(idx, n, multiset):
    if idx == len(inputList):
      return [multiset]

    if (len(inputList) - idx) * m > n:
      return []

    minNum = n if idx == len(inputList) - 1 else m
    maxNum = n - (len(inputList) - idx) * m + m

    results = []

    for i in xrange(minNum, maxNum + 1):
      multisetCopy = multiset.copy()
      multisetCopy[inputList[idx]] = i
      results = results + f(idx + 1, n - i, multisetCopy) 

    return results

  return f(0, k, {})

def distinct_permutations_from_multiset(multiset):
  def f(multiset, permutation):
    if multiset == {}:
      return [permutation]

    results = []

    for k in multiset:
      multisetCopy = multiset.copy()
      if multiset[k] == 1:
        del multisetCopy[k]
      else:
        multisetCopy[k] = multisetCopy[k] - 1
      results = results + f(multisetCopy, permutation + [k])

    return results

  return f(multiset, [])

输出：

print [distinct_permutations_from_multiset(x) for x in multisets_with_at_least_m([1,2,3], 5, 1)]

"""
 [[[1, 2, 3, 3, 3], [1, 3, 2, 3, 3], [1, 3, 3, 2, 3], [1, 3, 3, 3, 2]
 , [2, 1, 3, 3, 3], [2, 3, 1, 3, 3], [2, 3, 3, 1, 3], [2, 3, 3, 3, 1]
 , [3, 1, 2, 3, 3], [3, 1, 3, 2, 3], [3, 1, 3, 3, 2], [3, 2, 1, 3, 3]
 , [3, 2, 3, 1, 3], [3, 2, 3, 3, 1], [3, 3, 1, 2, 3], [3, 3, 1, 3, 2]
 , [3, 3, 2, 1, 3], [3, 3, 2, 3, 1], [3, 3, 3, 1, 2], [3, 3, 3, 2, 1]]
, [[1, 2, 2, 3, 3], [1, 2, 3, 2, 3], [1, 2, 3, 3, 2], [1, 3, 2, 2, 3]
 , [1, 3, 2, 3, 2], [1, 3, 3, 2, 2], [2, 1, 2, 3, 3], [2, 1, 3, 2, 3]
 , [2, 1, 3, 3, 2], [2, 2, 1, 3, 3], [2, 2, 3, 1, 3], [2, 2, 3, 3, 1]
 , [2, 3, 1, 2, 3], [2, 3, 1, 3, 2], [2, 3, 2, 1, 3], [2, 3, 2, 3, 1]
 , [2, 3, 3, 1, 2], [2, 3, 3, 2, 1], [3, 1, 2, 2, 3], [3, 1, 2, 3, 2]
 , [3, 1, 3, 2, 2], [3, 2, 1, 2, 3], [3, 2, 1, 3, 2], [3, 2, 2, 1, 3]
 , [3, 2, 2, 3, 1], [3, 2, 3, 1, 2], [3, 2, 3, 2, 1], [3, 3, 1, 2, 2]
 , [3, 3, 2, 1, 2], [3, 3, 2, 2, 1]]
, [[1, 2, 2, 2, 3], [1, 2, 2, 3, 2], [1, 2, 3, 2, 2], [1, 3, 2, 2, 2]
 , [2, 1, 2, 2, 3], [2, 1, 2, 3, 2], [2, 1, 3, 2, 2], [2, 2, 1, 2, 3]
 , [2, 2, 1, 3, 2], [2, 2, 2, 1, 3], [2, 2, 2, 3, 1], [2, 2, 3, 1, 2]
 , [2, 2, 3, 2, 1], [2, 3, 1, 2, 2], [2, 3, 2, 1, 2], [2, 3, 2, 2, 1]
 , [3, 1, 2, 2, 2], [3, 2, 1, 2, 2], [3, 2, 2, 1, 2], [3, 2, 2, 2, 1]]
, [[1, 1, 2, 3, 3], [1, 1, 3, 2, 3], [1, 1, 3, 3, 2], [1, 2, 1, 3, 3]
 , [1, 2, 3, 1, 3], [1, 2, 3, 3, 1], [1, 3, 1, 2, 3], [1, 3, 1, 3, 2]
 , [1, 3, 2, 1, 3], [1, 3, 2, 3, 1], [1, 3, 3, 1, 2], [1, 3, 3, 2, 1]
 , [2, 1, 1, 3, 3], [2, 1, 3, 1, 3], [2, 1, 3, 3, 1], [2, 3, 1, 1, 3]
 , [2, 3, 1, 3, 1], [2, 3, 3, 1, 1], [3, 1, 1, 2, 3], [3, 1, 1, 3, 2]
 , [3, 1, 2, 1, 3], [3, 1, 2, 3, 1], [3, 1, 3, 1, 2], [3, 1, 3, 2, 1]
 , [3, 2, 1, 1, 3], [3, 2, 1, 3, 1], [3, 2, 3, 1, 1], [3, 3, 1, 1, 2]
 , [3, 3, 1, 2, 1], [3, 3, 2, 1, 1]]
, [[1, 1, 2, 2, 3], [1, 1, 2, 3, 2], [1, 1, 3, 2, 2], [1, 2, 1, 2, 3]
 , [1, 2, 1, 3, 2], [1, 2, 2, 1, 3], [1, 2, 2, 3, 1], [1, 2, 3, 1, 2]
 , [1, 2, 3, 2, 1], [1, 3, 1, 2, 2], [1, 3, 2, 1, 2], [1, 3, 2, 2, 1]
 , [2, 1, 1, 2, 3], [2, 1, 1, 3, 2], [2, 1, 2, 1, 3], [2, 1, 2, 3, 1]
 , [2, 1, 3, 1, 2], [2, 1, 3, 2, 1], [2, 2, 1, 1, 3], [2, 2, 1, 3, 1]
 , [2, 2, 3, 1, 1], [2, 3, 1, 1, 2], [2, 3, 1, 2, 1], [2, 3, 2, 1, 1]
 , [3, 1, 1, 2, 2], [3, 1, 2, 1, 2], [3, 1, 2, 2, 1], [3, 2, 1, 1, 2]
 , [3, 2, 1, 2, 1], [3, 2, 2, 1, 1]]
, [[1, 1, 1, 2, 3], [1, 1, 1, 3, 2], [1, 1, 2, 1, 3], [1, 1, 2, 3, 1]
 , [1, 1, 3, 1, 2], [1, 1, 3, 2, 1], [1, 2, 1, 1, 3], [1, 2, 1, 3, 1]
 , [1, 2, 3, 1, 1], [1, 3, 1, 1, 2], [1, 3, 1, 2, 1], [1, 3, 2, 1, 1]
 , [2, 1, 1, 1, 3], [2, 1, 1, 3, 1], [2, 1, 3, 1, 1], [2, 3, 1, 1, 1]
 , [3, 1, 1, 1, 2], [3, 1, 1, 2, 1], [3, 1, 2, 1, 1], [3, 2, 1, 1, 1]]]
"""