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# 9.7. `itertools` — 为高效循环而创建迭代器的函数

2.3 新版功能.

本模块实现一系列 iterator ，这些迭代器受到APL，Haskell和SML的启发。为了适用于Python，它们都被重新写过。

本模块标准化了一个快速、高效利用内存的核心工具集，这些工具本身或组合都很有用。它们一起形成了“迭代器代数”，这使得在纯Python中有可能创建简洁又高效的专用工具。

For instance, SML provides a tabulation tool: `tabulate(f)` which produces a sequence `f(0), f(1), ...`. The same effect can be achieved in Python by combining `imap()` and `count()` to form `imap(f, count())`.

These tools and their built-in counterparts also work well with the high-speed functions in the `operator` module. For example, the multiplication operator can be mapped across two vectors to form an efficient dot-product: `sum(imap(operator.mul, vector1, vector2))`.

Infinite Iterators:

迭代器 | 实参 | 结果 | 示例
--- | --- | --- | ---
`count()` | start, [step] | start, start+step, start+2*step, … | `count(10) --> 10 11 12 13 14 ...`
`cycle()` | p | p0, p1, … plast, p0, p1, … | `cycle('ABCD') --> A B C D A B C D ...`
`repeat()` | elem [,n] | elem, elem, elem, … 重复无限次或n次 | `repeat(10, 3) --> 10 10 10`

根据最短输入序列长度停止的迭代器：

迭代器 | 实参 | 结果 | 示例
--- | --- | --- | ---
`chain()` | p, q, … | p0, p1, … plast, q0, q1, … | `chain('ABC', 'DEF') --> A B C D E F`
`compress()` | data, selectors | (d[0] if s[0]), (d[1] if s[1]), … | `compress('ABCDEF', [1,0,1,0,1,1]) --> A C E F`
`dropwhile()` | pred, seq | seq[n], seq[n+1], … 从pred首次真值测试失败开始 | `dropwhile(lambda x: x<5, [1,4,6,4,1]) --> 6 4 1`
`groupby()` | iterable[, keyfunc] | sub-iterators grouped by value of keyfunc(v) |
`ifilter()` | pred, seq | elements of seq where pred(elem) is true | `ifilter(lambda x: x%2, range(10)) --> 1 3 5 7 9`
`ifilterfalse()` | pred, seq | seq中pred(x)为假值的元素，x是seq中的元素。 | `ifilterfalse(lambda x: x%2, range(10)) --> 0 2 4 6 8`
`islice()` | seq, [start,] stop [, step] | seq[start:stop:step]中的元素 | `islice('ABCDEFG', 2, None) --> C D E F G`
`imap()` | func, p, q, … | func(p0, q0), func(p1, q1), … | `imap(pow, (2,3,10), (5,2,3)) --> 32 9 1000`
`starmap()` | func, seq | func(*seq[0]), func(*seq[1]), … | `starmap(pow, [(2,5), (3,2), (10,3)]) --> 32 9 1000`
`tee()` | it, n | it1, it2, … itn 将一个迭代器拆分为n个迭代器 |
`takewhile()` | pred, seq | seq[0], seq[1], …, 直到pred真值测试失败 | `takewhile(lambda x: x<5, [1,4,6,4,1]) --> 1 4`
`izip()` | p, q, … | (p[0], q[0]), (p[1], q[1]), … | `izip('ABCD', 'xy') --> Ax By`
`izip_longest()` | p, q, … | (p[0], q[0]), (p[1], q[1]), … | `izip_longest('ABCD', 'xy', fillvalue='-') --> Ax By C- D-`

Combinatoric generators:

迭代器 | 实参 | 结果
--- | --- | ---
`product()` | p, q, … [repeat=1] | 笛卡尔积，相当于嵌套的for循环
`permutations()` | p[, r] | 长度r元组，所有可能的排列，无重复元素
`combinations()` | p, r | 长度r元组，有序，无重复元素
`combinations_with_replacement()` | p, r | 长度r元组，有序，元素可重复
`product('ABCD', repeat=2)` |  | `AA AB AC AD BA BB BC BD CA CB CC CD DA DB DC DD`
`permutations('ABCD', 2)` |  | `AB AC AD BA BC BD CA CB CD DA DB DC`
`combinations('ABCD', 2)` |  | `AB AC AD BC BD CD`
`combinations_with_replacement('ABCD', 2)` |  | `AA AB AC AD BB BC BD CC CD DD`

## 9.7.1. Itertool函数

下列模块函数均创建并返回迭代器。有些迭代器不限制输出流长度，所以它们只应在能截断输出流的函数或循环中使用。

`itertools.``chain`(*iterables)

  创建一个迭代器，它首先返回第一个可迭代对象中所有元素，接着返回下一个可迭代对象中所有元素，直到耗尽所有可迭代对象中的元素。可将多个序列处理为单个序列。大致相当于：

      def chain(*iterables):
          # chain('ABC', 'DEF') --> A B C D E F
          for it in iterables:
              for element in it:
                  yield element

classmethod `chain.``from_iterable`(iterable)

  构建类似 `chain()` 迭代器的另一个选择。从一个单独的可迭代参数中得到链式输入，该参数是延迟计算的。大致相当于：

      def from_iterable(iterables):
          # chain.from_iterable(['ABC', 'DEF']) --> A B C D E F
          for it in iterables:
              for element in it:
                  yield element

  2.6 新版功能.

`itertools.``combinations`(iterable, r)

  返回由输入 iterable 中元素组成长度为 r 的子序列。

  组合按照字典序返回。所以如果输入 iterable 是有序的，生成的组合元组也是有序的。

  即使元素的值相同，不同位置的元素也被认为是不同的。如果元素各自不同，那么每个组合中没有重复元素。

  大致相当于：

      def combinations(iterable, r):
          # combinations('ABCD', 2) --> AB AC AD BC BD CD
          # combinations(range(4), 3) --> 012 013 023 123
          pool = tuple(iterable)
          n = len(pool)
          if r > n:
              return
          indices = range(r)
          yield tuple(pool[i] for i in indices)
          while True:
              for i in reversed(range(r)):
                  if indices[i] != i + n - r:
                      break
              else:
                  return
              indices[i] += 1
              for j in range(i+1, r):
                  indices[j] = indices[j-1] + 1
              yield tuple(pool[i] for i in indices)

  `combinations()` 的代码可被改写为 `permutations()` 过滤后的子序列，（相对于元素在输入中的位置）元素不是有序的。

      def combinations(iterable, r):
          pool = tuple(iterable)
          n = len(pool)
          for indices in permutations(range(n), r):
              if sorted(indices) == list(indices):
                  yield tuple(pool[i] for i in indices)

  当 `0 <= r <= n` 时，返回项的个数是 `n! / r! / (n-r)!`；当 `r > n` 时，返回项个数为0。

  2.6 新版功能.

`itertools.``combinations_with_replacement`(iterable, r)

  返回由输入 iterable 中元素组成的长度为 r 的子序列，允许每个元素可重复出现。

  组合按照字典序返回。所以如果输入 iterable 是有序的，生成的组合元组也是有序的。

  不同位置的元素是不同的，即使它们的值相同。因此如果输入中的元素都是不同的话，返回的组合中元素也都会不同。

  大致相当于：

      def combinations_with_replacement(iterable, r):
          # combinations_with_replacement('ABC', 2) --> AA AB AC BB BC CC
          pool = tuple(iterable)
          n = len(pool)
          if not n and r:
              return
          indices = [0] * r
          yield tuple(pool[i] for i in indices)
          while True:
              for i in reversed(range(r)):
                  if indices[i] != n - 1:
                      break
              else:
                  return
              indices[i:] = [indices[i] + 1] * (r - i)
              yield tuple(pool[i] for i in indices)

  `combinations_with_replacement()` 的代码可被改写为 `production()` 过滤后的子序列，（相对于元素在输入中的位置）元素不是有序的。

      def combinations_with_replacement(iterable, r):
          pool = tuple(iterable)
          n = len(pool)
          for indices in product(range(n), repeat=r):
              if sorted(indices) == list(indices):
                  yield tuple(pool[i] for i in indices)

  当 `n > 0` 时，返回项个数为 `(n+r-1)! / r! / (n-1)!`.

  2.7 新版功能.

`itertools.``compress`(data, selectors)

  创建一个迭代器，它返回 data 中经 selectors 真值测试为 `True` 的元素。迭代器在两者较短的长度处停止。大致相当于：

      def compress(data, selectors):
          # compress('ABCDEF', [1,0,1,0,1,1]) --> A C E F
          return (d for d, s in izip(data, selectors) if s)

  2.7 新版功能.

`itertools.``count`(start=0, step=1)

  Make an iterator that returns evenly spaced values starting with n. Often used as an argument to `imap()` to generate consecutive data points. Also, used with `izip()` to add sequence numbers. Equivalent to:

      def count(start=0, step=1):
          # count(10) --> 10 11 12 13 14 ...
          # count(2.5, 0.5) -> 2.5 3.0 3.5 ...
          n = start
          while True:
              yield n
              n += step

  当对浮点数计数时，替换为乘法代码有时精度会更好，例如： `(start + step * i for i in count())` 。

  在 2.7 版更改: added step argument and allowed non-integer arguments.

`itertools.``cycle`(iterable)

  创建一个迭代器，返回 iterable 中所有元素并保存一个副本。当取完 iterable 中所有元素，返回副本中的所有元素。无限重复。大致相当于：

      def cycle(iterable):
          # cycle('ABCD') --> A B C D A B C D A B C D ...
          saved = []
          for element in iterable:
              yield element
              saved.append(element)
          while saved:
              for element in saved:
                    yield element

  注意，该函数可能需要相当大的辅助空间（取决于 iterable 的长度）。

`itertools.``dropwhile`(predicate, iterable)

  创建一个迭代器，如果 predicate 为true，迭代器丢弃这些元素，然后返回其他元素。注意，迭代器在 predicate 首次为false之前不会产生任何输出，所以可能需要一定长度的启动时间。大致相当于：

      def dropwhile(predicate, iterable):
          # dropwhile(lambda x: x<5, [1,4,6,4,1]) --> 6 4 1
          iterable = iter(iterable)
          for x in iterable:
              if not predicate(x):
                  yield x
                  break
          for x in iterable:
              yield x

`itertools.``groupby`(iterable[, key])

  创建一个迭代器，返回 iterable 中连续的键和组。key 是一个计算元素键值函数。如果未指定或为 `None`，key 缺省为恒等函数（identity function），返回元素不变。一般来说，iterable 需用同一个键值函数预先排序。

  `groupby()` 操作类似于Unix中的 `uniq`。当每次 key 函数产生的键值改变时，迭代器会分组或生成一个新组（这就是为什么通常需要使用同一个键值函数先对数据进行排序）。这种行为与SQL的GROUP BY操作不同，SQL的操作会忽略输入的顺序将相同键值的元素分在同组中。

  返回的组本身也是一个迭代器，它与 `groupby()` 共享底层的可迭代对象。因为源是共享的，当 `groupby()` 对象向后迭代时，前一个组将消失。因此如果稍后还需要返回结果，可保存为列表：

      groups = []
      uniquekeys = []
      data = sorted(data, key=keyfunc)
      for k, g in groupby(data, keyfunc):
          groups.append(list(g))      # Store group iterator as a list
          uniquekeys.append(k)

  `groupby()` 大致相当于：

      class groupby(object):
          # [k for k, g in groupby('AAAABBBCCDAABBB')] --> A B C D A B
          # [list(g) for k, g in groupby('AAAABBBCCD')] --> AAAA BBB CC D
          def __init__(self, iterable, key=None):
              if key is None:
                  key = lambda x: x
              self.keyfunc = key
              self.it = iter(iterable)
              self.tgtkey = self.currkey = self.currvalue = object()
          def __iter__(self):
              return self
          def next(self):
              while self.currkey == self.tgtkey:
                  self.currvalue = next(self.it)    # Exit on StopIteration
                  self.currkey = self.keyfunc(self.currvalue)
              self.tgtkey = self.currkey
              return (self.currkey, self._grouper(self.tgtkey))
          def _grouper(self, tgtkey):
              while self.currkey == tgtkey:
                  yield self.currvalue
                  self.currvalue = next(self.it)    # Exit on StopIteration
                  self.currkey = self.keyfunc(self.currvalue)

  2.4 新版功能.

`itertools.``ifilter`(predicate, iterable)

  Make an iterator that filters elements from iterable returning only those for which the predicate is `True`. If predicate is `None`, return the items that are true. Roughly equivalent to:

      def ifilter(predicate, iterable):
          # ifilter(lambda x: x%2, range(10)) --> 1 3 5 7 9
          if predicate is None:
              predicate = bool
          for x in iterable:
              if predicate(x):
                  yield x

`itertools.``ifilterfalse`(predicate, iterable)

  创建一个迭代器，只返回 iterable 中 predicate 为 `False` 的元素。如果 predicate 是 `None`，返回真值测试为false的元素。大致相当于：

      def ifilterfalse(predicate, iterable):
          # ifilterfalse(lambda x: x%2, range(10)) --> 0 2 4 6 8
          if predicate is None:
              predicate = bool
          for x in iterable:
              if not predicate(x):
                  yield x

`itertools.``imap`(function, *iterables)

  Make an iterator that computes the function using arguments from each of the iterables. If function is set to `None`, then `imap()` returns the arguments as a tuple. Like `map()` but stops when the shortest iterable is exhausted instead of filling in `None` for shorter iterables. The reason for the difference is that infinite iterator arguments are typically an error for `map()` (because the output is fully evaluated) but represent a common and useful way of supplying arguments to `imap()`. Roughly equivalent to:

      def imap(function, *iterables):
          # imap(pow, (2,3,10), (5,2,3)) --> 32 9 1000
          iterables = map(iter, iterables)
          while True:
              args = [next(it) for it in iterables]
              if function is None:
                  yield tuple(args)
              else:
                  yield function(*args)

`itertools.``islice`(iterable, stop)
`itertools.``islice`(iterable, start, stop[, step])

  创建一个迭代器，返回从 iterable 里选中的元素。如果 start 不是0，跳过 iterable 中的元素，直到到达 start 这个位置。之后迭代器连续返回元素，除非 step 设置的值很高导致被跳过。如果 stop 为 `None`，迭代器耗光为止；否则，在指定的位置停止。与普通的切片不同，`islice()` 不支持将 start ， stop ，或 step 设为负值。可用来从内部数据结构被压平的数据中提取相关字段（例如一个多行报告，它的名称字段出现在每三行上）。大致相当于：

      def islice(iterable, *args):
          # islice('ABCDEFG', 2) --> A B
          # islice('ABCDEFG', 2, 4) --> C D
          # islice('ABCDEFG', 2, None) --> C D E F G
          # islice('ABCDEFG', 0, None, 2) --> A C E G
          s = slice(*args)
          start, stop, step = s.start or 0, s.stop or sys.maxint, s.step or 1
          it = iter(xrange(start, stop, step)))
          try:
              nexti = next(it)
          except StopIteration:
              # Consume *iterable* up to the *start* position.
              for i, element in izip(xrange(start), iterable):
                  pass
              return
          try:
              for i, element in enumerate(iterable):
                  if i == nexti:
                      yield element
                      nexti = next(it)
          except StopIteration:
              # Consume to *stop*.
              for i, element in izip(xrange(i + 1, stop), iterable):
                  pass

  如果 start 为 `None`，迭代从0开始。如果 step 为 `None` ，步长缺省为1。

  在 2.5 版更改: accept `None` values for default start and step.

`itertools.``izip`(*iterables)

  Make an iterator that aggregates elements from each of the iterables. Like `zip()` except that it returns an iterator instead of a list. Used for lock-step iteration over several iterables at a time. Roughly equivalent to:

      def izip(*iterables):
          # izip('ABCD', 'xy') --> Ax By
          iterators = map(iter, iterables)
          while iterators:
              yield tuple(map(next, iterators))

  在 2.4 版更改: When no iterables are specified, returns a zero length iterator instead of raising a `TypeError` exception.

  The left-to-right evaluation order of the iterables is guaranteed. This makes possible an idiom for clustering a data series into n-length groups using `izip(*[iter(s)]*n)`.

  `izip()` should only be used with unequal length inputs when you don’t care about trailing, unmatched values from the longer iterables. If those values are important, use `izip_longest()` instead.

`itertools.``izip_longest`(*iterables[, fillvalue])

  创建一个迭代器，从每个可迭代对象中收集元素。如果可迭代对象的长度未对齐，将根据 fillvalue 填充缺失值。迭代持续到耗光最长的可迭代对象。大致相当于：

      class ZipExhausted(Exception):
          pass

      def izip_longest(*args, **kwds):
          # izip_longest('ABCD', 'xy', fillvalue='-') --> Ax By C- D-
          fillvalue = kwds.get('fillvalue')
          counter = [len(args) - 1]
          def sentinel():
              if not counter[0]:
                  raise ZipExhausted
              counter[0] -= 1
              yield fillvalue
          fillers = repeat(fillvalue)
          iterators = [chain(it, sentinel(), fillers) for it in args]
          try:
              while iterators:
                  yield tuple(map(next, iterators))
          except ZipExhausted:
              pass

  If one of the iterables is potentially infinite, then the `izip_longest()` function should be wrapped with something that limits the number of calls (for example `islice()` or `takewhile()`). If not specified, fillvalue defaults to `None`.

  2.6 新版功能.

`itertools.``permutations`(iterable[, r])

  连续返回由 iterable 元素生成长度为 r 的排列。

  如果 r 未指定或为 `None` ，r 默认设置为 iterable 的长度，这种情况下，生成所有全长排列。

  排列依字典序发出。因此，如果 iterable 是已排序的，排列元组将有序地产出。

  即使元素的值相同，不同位置的元素也被认为是不同的。如果元素值都不同，每个排列中的元素值不会重复。

  大致相当于：

      def permutations(iterable, r=None):
          # permutations('ABCD', 2) --> AB AC AD BA BC BD CA CB CD DA DB DC
          # permutations(range(3)) --> 012 021 102 120 201 210
          pool = tuple(iterable)
          n = len(pool)
          r = n if r is None else r
          if r > n:
              return
          indices = range(n)
          cycles = range(n, n-r, -1)
          yield tuple(pool[i] for i in indices[:r])
          while n:
              for i in reversed(range(r)):
                  cycles[i] -= 1
                  if cycles[i] == 0:
                      indices[i:] = indices[i+1:] + indices[i:i+1]
                      cycles[i] = n - i
                  else:
                      j = cycles[i]
                      indices[i], indices[-j] = indices[-j], indices[i]
                      yield tuple(pool[i] for i in indices[:r])
                      break
              else:
                  return

  `permutations()` 的代码也可被改写为 `product()` 的子序列，只要将含有重复元素（来自输入中同一位置的）的项排除。

      def permutations(iterable, r=None):
          pool = tuple(iterable)
          n = len(pool)
          r = n if r is None else r
          for indices in product(range(n), repeat=r):
              if len(set(indices)) == r:
                  yield tuple(pool[i] for i in indices)

  当 `0 <= r <= n` ，返回项个数为 `n! / (n-r)!` ；当 `r > n` ，返回项个数为0。

  2.6 新版功能.

`itertools.``product`(*iterables[, repeat])

  可迭代对象输入的笛卡儿积。

  大致相当于生成器表达式中的嵌套循环。例如， `product(A, B)` 和 `((x,y) for x in A for y in B)` 返回结果一样。

  嵌套循环像里程表那样循环变动，每次迭代时将最右侧的元素向后迭代。这种模式形成了一种字典序，因此如果输入的可迭代对象是已排序的，笛卡尔积元组依次序发出。

  要计算可迭代对象自身的笛卡尔积，将可选参数 repeat 设定为要重复的次数。例如，`product(A, repeat=4)` 和 `product(A, A, A, A)` 是一样的。

  该函数大致相当于下面的代码，只不过实际实现方案不会在内存中创建中间结果。

      def product(*args, **kwds):
          # product('ABCD', 'xy') --> Ax Ay Bx By Cx Cy Dx Dy
          # product(range(2), repeat=3) --> 000 001 010 011 100 101 110 111
          pools = map(tuple, args) * kwds.get('repeat', 1)
          result = [[]]
          for pool in pools:
              result = [x+[y] for x in result for y in pool]
          for prod in result:
              yield tuple(prod)

  2.6 新版功能.

`itertools.``repeat`(object[, times])

  Make an iterator that returns object over and over again. Runs indefinitely unless the times argument is specified. Used as argument to `imap()` for invariant function parameters. Also used with `izip()` to create constant fields in a tuple record. Roughly equivalent to:

      def repeat(object, times=None):
          # repeat(10, 3) --> 10 10 10
          if times is None:
              while True:
                  yield object
          else:
              for i in xrange(times):
                  yield object

  A common use for repeat is to supply a stream of constant values to imap or zip:

      >>> list(imap(pow, xrange(10), repeat(2)))
      [0, 1, 4, 9, 16, 25, 36, 49, 64, 81]

`itertools.``starmap`(function, iterable)

  Make an iterator that computes the function using arguments obtained from the iterable. Used instead of `imap()` when argument parameters are already grouped in tuples from a single iterable (the data has been “pre-zipped”). The difference between `imap()` and `starmap()` parallels the distinction between `function(a,b)` and `function(*c)`. Roughly equivalent to:

      def starmap(function, iterable):
          # starmap(pow, [(2,5), (3,2), (10,3)]) --> 32 9 1000
          for args in iterable:
              yield function(*args)

  在 2.6 版更改: Previously, `starmap()` required the function arguments to be tuples. Now, any iterable is allowed.

`itertools.``takewhile`(predicate, iterable)

  创建一个迭代器，只要 predicate 为真就从可迭代对象中返回元素。大致相当于:

      def takewhile(predicate, iterable):
          # takewhile(lambda x: x<5, [1,4,6,4,1]) --> 1 4
          for x in iterable:
              if predicate(x):
                  yield x
              else:
                  break

`itertools.``tee`(iterable[, n=2])

  Return n independent iterators from a single iterable. Roughly equivalent to:

      def tee(iterable, n=2):
          it = iter(iterable)
          deques = [collections.deque() for i in range(n)]
          def gen(mydeque):
              while True:
                  if not mydeque:             # when the local deque is empty
                      newval = next(it)       # fetch a new value and
                      for d in deques:        # load it to all the deques
                          d.append(newval)
                  yield mydeque.popleft()
          return tuple(gen(d) for d in deques)

  一旦 `tee()` 实施了一次分裂，原有的 iterable 不应再被使用；否则tee对象无法得知 iterable 可能已向后迭代。

  `tee` 迭代器不是线程安全的。当同时使用由同一个 `tee()` 调用所返回的迭代器时可能引发 `RuntimeError`，即使原本的 iterable 是线程安全的。

  该迭代工具可能需要相当大的辅助存储空间（这取决于要保存多少临时数据）。通常，如果一个迭代器在另一个迭代器开始之前就要使用大部份或全部数据，使用 `list()` 会比 `tee()` 更快。

  2.4 新版功能.

## 9.7.2. Recipes

本节将展示如何使用现有的itertools作为基础构件来创建扩展的工具集。

扩展的工具提供了与底层工具集相同的高性能。保持了超棒的内存利用率，因为一次只处理一个元素，而不是将整个可迭代对象加载到内存。代码量保持得很小，以函数式风格将这些工具连接在一起，有助于消除临时变量。速度依然很快，因为倾向于使用“矢量化”构件来取代解释器开销大的 for 循环和 generator 。

    def take(n, iterable):
        "Return first n items of the iterable as a list"
        return list(islice(iterable, n))

    def tabulate(function, start=0):
        "Return function(0), function(1), ..."
        return imap(function, count(start))

    def consume(iterator, n=None):
        "Advance the iterator n-steps ahead. If n is None, consume entirely."
        # Use functions that consume iterators at C speed.
        if n is None:
            # feed the entire iterator into a zero-length deque
            collections.deque(iterator, maxlen=0)
        else:
            # advance to the empty slice starting at position n
            next(islice(iterator, n, n), None)

    def nth(iterable, n, default=None):
        "Returns the nth item or a default value"
        return next(islice(iterable, n, None), default)

    def all_equal(iterable):
        "Returns True if all the elements are equal to each other"
        g = groupby(iterable)
        return next(g, True) and not next(g, False)

    def quantify(iterable, pred=bool):
        "Count how many times the predicate is true"
        return sum(imap(pred, iterable))

    def padnone(iterable):
        """Returns the sequence elements and then returns None indefinitely.

        Useful for emulating the behavior of the built-in map() function.
        """
        return chain(iterable, repeat(None))

    def ncycles(iterable, n):
        "Returns the sequence elements n times"
        return chain.from_iterable(repeat(tuple(iterable), n))

    def dotproduct(vec1, vec2):
        return sum(imap(operator.mul, vec1, vec2))

    def flatten(listOfLists):
        "Flatten one level of nesting"
        return chain.from_iterable(listOfLists)

    def repeatfunc(func, times=None, *args):
        """Repeat calls to func with specified arguments.

        Example:  repeatfunc(random.random)
        """
        if times is None:
            return starmap(func, repeat(args))
        return starmap(func, repeat(args, times))

    def pairwise(iterable):
        "s -> (s0,s1), (s1,s2), (s2, s3), ..."
        a, b = tee(iterable)
        next(b, None)
        return izip(a, b)

    def grouper(iterable, n, fillvalue=None):
        "Collect data into fixed-length chunks or blocks"
        # grouper('ABCDEFG', 3, 'x') --> ABC DEF Gxx
        args = [iter(iterable)] * n
        return izip_longest(fillvalue=fillvalue, *args)

    def roundrobin(*iterables):
        "roundrobin('ABC', 'D', 'EF') --> A D E B F C"
        # Recipe credited to George Sakkis
        pending = len(iterables)
        nexts = cycle(iter(it).next for it in iterables)
        while pending:
            try:
                for next in nexts:
                    yield next()
            except StopIteration:
                pending -= 1
                nexts = cycle(islice(nexts, pending))

    def powerset(iterable):
        "powerset([1,2,3]) --> () (1,) (2,) (3,) (1,2) (1,3) (2,3) (1,2,3)"
        s = list(iterable)
        return chain.from_iterable(combinations(s, r) for r in range(len(s)+1))

    def unique_everseen(iterable, key=None):
        "List unique elements, preserving order. Remember all elements ever seen."
        # unique_everseen('AAAABBBCCDAABBB') --> A B C D
        # unique_everseen('ABBCcAD', str.lower) --> A B C D
        seen = set()
        seen_add = seen.add
        if key is None:
            for element in ifilterfalse(seen.__contains__, iterable):
                seen_add(element)
                yield element
        else:
            for element in iterable:
                k = key(element)
                if k not in seen:
                    seen_add(k)
                    yield element

    def unique_justseen(iterable, key=None):
        "List unique elements, preserving order. Remember only the element just seen."
        # unique_justseen('AAAABBBCCDAABBB') --> A B C D A B
        # unique_justseen('ABBCcAD', str.lower) --> A B C A D
        return imap(next, imap(itemgetter(1), groupby(iterable, key)))

    def iter_except(func, exception, first=None):
        """ Call a function repeatedly until an exception is raised.

        Converts a call-until-exception interface to an iterator interface.
        Like __builtin__.iter(func, sentinel) but uses an exception instead
        of a sentinel to end the loop.

        Examples:
            bsddbiter = iter_except(db.next, bsddb.error, db.first)
            heapiter = iter_except(functools.partial(heappop, h), IndexError)
            dictiter = iter_except(d.popitem, KeyError)
            dequeiter = iter_except(d.popleft, IndexError)
            queueiter = iter_except(q.get_nowait, Queue.Empty)
            setiter = iter_except(s.pop, KeyError)

        """
        try:
            if first is not None:
                yield first()
            while 1:
                yield func()
        except exception:
            pass

    def random_product(*args, **kwds):
        "Random selection from itertools.product(*args, **kwds)"
        pools = map(tuple, args) * kwds.get('repeat', 1)
        return tuple(random.choice(pool) for pool in pools)

    def random_permutation(iterable, r=None):
        "Random selection from itertools.permutations(iterable, r)"
        pool = tuple(iterable)
        r = len(pool) if r is None else r
        return tuple(random.sample(pool, r))

    def random_combination(iterable, r):
        "Random selection from itertools.combinations(iterable, r)"
        pool = tuple(iterable)
        n = len(pool)
        indices = sorted(random.sample(xrange(n), r))
        return tuple(pool[i] for i in indices)

    def random_combination_with_replacement(iterable, r):
        "Random selection from itertools.combinations_with_replacement(iterable, r)"
        pool = tuple(iterable)
        n = len(pool)
        indices = sorted(random.randrange(n) for i in xrange(r))
        return tuple(pool[i] for i in indices)

    def tee_lookahead(t, i):
        """Inspect the i-th upcomping value from a tee object
           while leaving the tee object at its current position.

           Raise an IndexError if the underlying iterator doesn't
           have enough values.

        """
        for value in islice(t.__copy__(), i, None):
            return value
        raise IndexError(i)

注意，通过将全局查找替换为局部变量的缺省值，上述配方中有很多可以这样优化。例如， dotproduct 配方可以这样写：

    def dotproduct(vec1, vec2, sum=sum, imap=imap, mul=operator.mul):
        return sum(imap(mul, vec1, vec2))