Python 中的快速组合生成器

Question

作为 python 中一个大型项目的一部分，我需要一个快速生成器函数来生成小于 n 的所有可能的非负整数集，这样每个集最多有s个元素且集合中最大数与最小数之差小于w.

到目前为止我实现的最快的实现是使用 itertools:

import itertools

def subsample(n, s, w):
    nn = range(w)
    for p in range(s):
        o = list(itertools.combinations(nn, p+1))
        for t in o:
            yield t
        for _ in range(0, n-w):
            pt = o
            o = [tuple([op + 1 for op in list(u)]) for u in pt]
            for t in list((set(o) ^ set(pt)) & set(o)):
                yield t

例如：

In [1]: list(subsample(6,3,3))
Out [1]: [(0,), (1,), (2,), (3,), (4,), (5,), (0, 1), (0, 2), (1, 2), (1, 3), (2, 3), (3, 4), (2, 4), (4, 5), (3, 5), (0, 1, 2), (1, 2, 3), (2, 3, 4), (3, 4, 5)]

但我相信一定有更有效的方法来做到这一点。有什么可以让它更快的吗？

Answer 1

以下是一些基于 Knuth 算法的生成器。

subsets4 生成 1..n 的所有子集 k 或更少的元素
subsets5 将 subsets4 生成的子集限制为最大差异为 w
subsets 生成长度为 k

1..n

subsets2 将 subsets 生成的子集限制为最大差异为 w

运行 megatest() 函数测试 subsets5 生成器的多个值 n, k 和 w.

# all subsets of k or less elements of 1..n
def subsets4(n,k):
  a = [ 0 ] * k
  i = 0
  while i >= 0:
    a[i] += 1
    yield a[0:i+1]
    r = a[i]+1
    i += 1
    while i < k and r <= n:
      a[i] = r
      yield a[0:i+1]
      i += 1
      r += 1
    i -= 1
    if a[i] >= n:
      i -= 1

# all subsets of k or less elements of 1..n with max difference <= w
def subsets5(n,k,w):
  a = [ 0 ] * k
  i = 0
  while i >= 0:
    a[i] += 1
    yield a[0:i+1]
    r = a[i]+1
    i += 1
    while i < k and r <= n and r-a[0] <= w:
      a[i] = r
      yield a[0:i+1]
      i += 1
      r += 1
    i -= 1
    if a[i] >= n or a[i]+1-a[0] > w:
      i -= 1

# all subsets of 1..n having exactly k elements
def subsets(n,k):
  a = range(1,k+1)
  while a[0] <= n+1-k:
    yield a
    # find i
    i = k-1
    while i >= 0 and a[i]+k-i >= n+1: i -= 1
    r = a[i]
    a[i] += 1
    j = 2
    i += 1
    while i < k:
      a[i] = r + j
      i += 1
      j += 1

# all subsets of 1..n having exactly k elements and whose max
# difference is w
def subsets2(n,k,w):
  if k > w: return
  a = range(1,k+1)
  while a[0] <= n+1-k:
    yield a
    i = k-1
    while i >= 0 and (a[i]+k-i >= n+1 or a[i]+k-i-a[0] > w) : i -= 1
    r = a[i]
    a[i] += 1
    j = 2
    i += 1
    while i < k:
      a[i] = r + j
      i += 1
      j += 1

def test(n,k,w):
  s1 = [ s for s in subsets4(n,k) if s[-1] - s[0] <= w ]
  s2 = [ s for s in subsets5(n,k,w) ]
  if s1 == s2:
    print "OK", n,k,w
    return 0
  else:
    print "NOT OK", n, k, w
    return 1 

# for s in subsets2(10,3,4): print s
# for s in subsets(10,3): print s
def megatest():
  failed = 0
  for n in xrange(10,20):
    for k in xrange(1,n+1):
      for w in xrange(k,n+1):
        failed += test(n,k,w)
  print "failed:", failed

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