Doc/library/itertools.rst

:mod:`itertools` --- Functions creating iterators for efficient looping
=======================================================================

.. module:: itertools
   :synopsis: Functions creating iterators for efficient looping.
.. moduleauthor:: Raymond Hettinger <python@rcn.com>
.. sectionauthor:: Raymond Hettinger <python@rcn.com>


.. testsetup::

   from itertools import *


This module implements a number of :term:`iterator` building blocks inspired
by constructs from APL, Haskell, and SML.  Each has been recast in a form
suitable for Python.

The module standardizes a core set of fast, memory efficient tools that are
useful by themselves or in combination.  Together, they form an "iterator
algebra" making it possible to construct specialized tools succinctly and
efficiently in pure Python.

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 :func:`map` and :func:`count` to form ``map(f, count())``.

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


**Infinite Iterators:**

==================  =================       =================================================               =========================================
Iterator            Arguments               Results                                                         Example
==================  =================       =================================================               =========================================
:func:`count`       start, [step]           start, start+step, start+2*step, ...                            ``count(10) --> 10 11 12 13 14 ...``
:func:`cycle`       p                       p0, p1, ... plast, p0, p1, ...                                  ``cycle('ABCD') --> A B C D A B C D ...``
:func:`repeat`      elem [,n]               elem, elem, elem, ... endlessly or up to n times                ``repeat(10, 3) --> 10 10 10``
==================  =================       =================================================               =========================================

**Iterators terminating on the shortest input sequence:**

====================    ============================    =================================================   =============================================================
Iterator                Arguments                       Results                                             Example
====================    ============================    =================================================   =============================================================
:func:`accumulate`      p [,func]                       p0, p0+p1, p0+p1+p2, ...                            ``accumulate([1,2,3,4,5]) --> 1 3 6 10 15``
:func:`chain`           p, q, ...                       p0, p1, ... plast, q0, q1, ...                      ``chain('ABC', 'DEF') --> A B C D E F``
:func:`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``
:func:`dropwhile`       pred, seq                       seq[n], seq[n+1], starting when pred fails          ``dropwhile(lambda x: x<5, [1,4,6,4,1]) --> 6 4 1``
:func:`filterfalse`     pred, seq                       elements of seq where pred(elem) is False           ``filterfalse(lambda x: x%2, range(10)) --> 0 2 4 6 8``
:func:`groupby`         iterable[, keyfunc]             sub-iterators grouped by value of keyfunc(v)
:func:`islice`          seq, [start,] stop [, step]     elements from seq[start:stop:step]                  ``islice('ABCDEFG', 2, None) --> C D E F G``
:func:`starmap`         func, seq                       func(\*seq[0]), func(\*seq[1]), ...                 ``starmap(pow, [(2,5), (3,2), (10,3)]) --> 32 9 1000``
:func:`takewhile`       pred, seq                       seq[0], seq[1], until pred fails                    ``takewhile(lambda x: x<5, [1,4,6,4,1]) --> 1 4``
:func:`tee`             it, n                           it1, it2 , ... itn  splits one iterator into n
:func:`zip_longest`     p, q, ...                       (p[0], q[0]), (p[1], q[1]), ...                     ``zip_longest('ABCD', 'xy', fillvalue='-') --> Ax By C- D-``
====================    ============================    =================================================   =============================================================

**Combinatoric generators:**

==============================================   ====================       =============================================================
Iterator                                         Arguments                  Results
==============================================   ====================       =============================================================
:func:`product`                                  p, q, ... [repeat=1]       cartesian product, equivalent to a nested for-loop
:func:`permutations`                             p[, r]                     r-length tuples, all possible orderings, no repeated elements
:func:`combinations`                             p, r                       r-length tuples, in sorted order, no repeated elements
:func:`combinations_with_replacement`            p, r                       r-length tuples, in sorted order, with repeated elements
``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``
==============================================   ====================       =============================================================


.. _itertools-functions:

Itertool functions
------------------

The following module functions all construct and return iterators. Some provide
streams of infinite length, so they should only be accessed by functions or
loops that truncate the stream.

.. function:: accumulate(iterable[, func])

    Make an iterator that returns accumulated sums. Elements may be any addable
    type including :class:`Decimal` or :class:`Fraction`.  If the optional
    *func* argument is supplied, it should be a function of two arguments
    and it will be used instead of addition.

    Equivalent to::

        def accumulate(iterable, func=operator.add):
            'Return running totals'
            # accumulate([1,2,3,4,5]) --> 1 3 6 10 15
            # accumulate([1,2,3,4,5], operator.mul) --> 1 2 6 24 120
            it = iter(iterable)
            total = next(it)
            yield total
            for element in it:
                total = func(total, element)
                yield total

    There are a number of uses for the *func* argument.  It can be set to
    :func:`min` for a running minimum, :func:`max` for a running maximum, or
    :func:`operator.mul` for a running product.  Amortization tables can be
    built by accumulating interest and applying payments.  First-order
    `recurrence relations <http://en.wikipedia.org/wiki/Recurrence_relation>`_
    can be modeled by supplying the initial value in the iterable and using only
    the accumulated total in *func* argument::

      >>> data = [3, 4, 6, 2, 1, 9, 0, 7, 5, 8]
      >>> list(accumulate(data, operator.mul))     # running product
      [3, 12, 72, 144, 144, 1296, 0, 0, 0, 0]
      >>> list(accumulate(data, max))              # running maximum
      [3, 4, 6, 6, 6, 9, 9, 9, 9, 9]

      # Amortize a 5% loan of 1000 with 4 annual payments of 90
      >>> cashflows = [1000, -90, -90, -90, -90]
      >>> list(accumulate(cashflows, lambda bal, pmt: bal*1.05 + pmt))
      [1000, 960.0, 918.0, 873.9000000000001, 827.5950000000001]

      # Chaotic recurrence relation http://en.wikipedia.org/wiki/Logistic_map
      >>> logistic_map = lambda x, _:  r * x * (1 - x)
      >>> r = 3.8
      >>> x0 = 0.4
      >>> inputs = repeat(x0, 36)     # only the initial value is used
      >>> [format(x, '.2f') for x in accumulate(inputs, logistic_map)]
      ['0.40', '0.91', '0.30', '0.81', '0.60', '0.92', '0.29', '0.79', '0.63',
       '0.88' ,'0.39', '0.90', '0.33', '0.84', '0.52', '0.95', '0.18', '0.57',
       '0.93', '0.25', '0.71', '0.79', '0.63', '0.88', '0.39', '0.91', '0.32',
       '0.83', '0.54', '0.95', '0.20', '0.60', '0.91', '0.30', '0.80', '0.60']

    .. versionadded:: 3.2

    .. versionchanged:: 3.3
       Added the optional *func* parameter.

.. function:: chain(*iterables)

   Make an iterator that returns elements from the first iterable until it is
   exhausted, then proceeds to the next iterable, until all of the iterables are
   exhausted.  Used for treating consecutive sequences as a single sequence.
   Equivalent to::

      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)

   Alternate constructor for :func:`chain`.  Gets chained inputs from a
   single iterable argument that is evaluated lazily.  Equivalent to::

      @classmethod
      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


.. function:: combinations(iterable, r)

   Return *r* length subsequences of elements from the input *iterable*.

   Combinations are emitted in lexicographic sort order.  So, if the
   input *iterable* is sorted, the combination tuples will be produced
   in sorted order.

   Elements are treated as unique based on their position, not on their
   value.  So if the input elements are unique, there will be no repeat
   values in each combination.

   Equivalent to::

        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 = list(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)

   The code for :func:`combinations` can be also expressed as a subsequence
   of :func:`permutations` after filtering entries where the elements are not
   in sorted order (according to their position in the input pool)::

        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)

   The number of items returned is ``n! / r! / (n-r)!`` when ``0 <= r <= n``
   or zero when ``r > n``.

.. function:: combinations_with_replacement(iterable, r)

   Return *r* length subsequences of elements from the input *iterable*
   allowing individual elements to be repeated more than once.

   Combinations are emitted in lexicographic sort order.  So, if the
   input *iterable* is sorted, the combination tuples will be produced
   in sorted order.

   Elements are treated as unique based on their position, not on their
   value.  So if the input elements are unique, the generated combinations
   will also be unique.

   Equivalent to::

        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)

   The code for :func:`combinations_with_replacement` can be also expressed as
   a subsequence of :func:`product` after filtering entries where the elements
   are not in sorted order (according to their position in the input pool)::

        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)

   The number of items returned is ``(n+r-1)! / r! / (n-1)!`` when ``n > 0``.

   .. versionadded:: 3.1


.. function:: compress(data, selectors)

   Make an iterator that filters elements from *data* returning only those that
   have a corresponding element in *selectors* that evaluates to ``True``.
   Stops when either the *data* or *selectors* iterables has been exhausted.
   Equivalent to::

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

   .. versionadded:: 3.1


.. function:: count(start=0, step=1)

   Make an iterator that returns evenly spaced values starting with *n*. Often
   used as an argument to :func:`map` to generate consecutive data points.
   Also, used with :func:`zip` 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

   When counting with floating point numbers, better accuracy can sometimes be
   achieved by substituting multiplicative code such as: ``(start + step * i
   for i in count())``.

   .. versionchanged:: 3.1
      Added *step* argument and allowed non-integer arguments.

.. function:: cycle(iterable)

   Make an iterator returning elements from the iterable and saving a copy of each.
   When the iterable is exhausted, return elements from the saved copy.  Repeats
   indefinitely.  Equivalent to::

      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

   Note, this member of the toolkit may require significant auxiliary storage
   (depending on the length of the iterable).


.. function:: dropwhile(predicate, iterable)

   Make an iterator that drops elements from the iterable as long as the predicate
   is true; afterwards, returns every element.  Note, the iterator does not produce
   *any* output until the predicate first becomes false, so it may have a lengthy
   start-up time.  Equivalent to::

      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

.. function:: filterfalse(predicate, iterable)

   Make an iterator that filters elements from iterable returning only those for
   which the predicate is ``False``. If *predicate* is ``None``, return the items
   that are false. Equivalent to::

      def filterfalse(predicate, iterable):
          # filterfalse(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


.. function:: groupby(iterable, key=None)

   Make an iterator that returns consecutive keys and groups from the *iterable*.
   The *key* is a function computing a key value for each element.  If not
   specified or is ``None``, *key* defaults to an identity function and returns
   the element unchanged.  Generally, the iterable needs to already be sorted on
   the same key function.

   The operation of :func:`groupby` is similar to the ``uniq`` filter in Unix.  It
   generates a break or new group every time the value of the key function changes
   (which is why it is usually necessary to have sorted the data using the same key
   function).  That behavior differs from SQL's GROUP BY which aggregates common
   elements regardless of their input order.

   The returned group is itself an iterator that shares the underlying iterable
   with :func:`groupby`.  Because the source is shared, when the :func:`groupby`
   object is advanced, the previous group is no longer visible.  So, if that data
   is needed later, it should be stored as a list::

      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)

   :func:`groupby` is equivalent to::

      class groupby:
          # [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)


.. function:: islice(iterable, [start,] stop [, step])

   Make an iterator that returns selected elements from the iterable. If *start* is
   non-zero, then elements from the iterable are skipped until start is reached.
   Afterward, elements are returned consecutively unless *step* is set higher than
   one which results in items being skipped.  If *stop* is ``None``, then iteration
   continues until the iterator is exhausted, if at all; otherwise, it stops at the
   specified position.  Unlike regular slicing, :func:`islice` does not support
   negative values for *start*, *stop*, or *step*.  Can be used to extract related
   fields from data where the internal structure has been flattened (for example, a
   multi-line report may list a name field on every third line).  Equivalent to::

      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)
          it = iter(range(s.start or 0, s.stop or sys.maxsize, s.step or 1))
          nexti = next(it)
          for i, element in enumerate(iterable):
              if i == nexti:
                  yield element
                  nexti = next(it)

   If *start* is ``None``, then iteration starts at zero. If *step* is ``None``,
   then the step defaults to one.


.. function:: permutations(iterable, r=None)

   Return successive *r* length permutations of elements in the *iterable*.

   If *r* is not specified or is ``None``, then *r* defaults to the length
   of the *iterable* and all possible full-length permutations
   are generated.

   Permutations are emitted in lexicographic sort order.  So, if the
   input *iterable* is sorted, the permutation tuples will be produced
   in sorted order.

   Elements are treated as unique based on their position, not on their
   value.  So if the input elements are unique, there will be no repeat
   values in each permutation.

   Equivalent to::

        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 = list(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

   The code for :func:`permutations` can be also expressed as a subsequence of
   :func:`product`, filtered to exclude entries with repeated elements (those
   from the same position in the input pool)::

        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)

   The number of items returned is ``n! / (n-r)!`` when ``0 <= r <= n``
   or zero when ``r > n``.

.. function:: product(*iterables, repeat=1)

   Cartesian product of input iterables.

   Equivalent to nested for-loops in a generator expression. For example,
   ``product(A, B)`` returns the same as ``((x,y) for x in A for y in B)``.

   The nested loops cycle like an odometer with the rightmost element advancing
   on every iteration.  This pattern creates a lexicographic ordering so that if
   the input's iterables are sorted, the product tuples are emitted in sorted
   order.

   To compute the product of an iterable with itself, specify the number of
   repetitions with the optional *repeat* keyword argument.  For example,
   ``product(A, repeat=4)`` means the same as ``product(A, A, A, A)``.

   This function is equivalent to the following code, except that the
   actual implementation does not build up intermediate results in memory::

       def product(*args, repeat=1):
           # 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 = [tuple(pool) for pool in args] * repeat
           result = [[]]
           for pool in pools:
               result = [x+[y] for x in result for y in pool]
           for prod in result:
               yield tuple(prod)


.. function:: 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 :func:`map` for
   invariant parameters to the called function.  Also used with :func:`zip` to
   create an invariant part of a tuple record.  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 range(times):
                  yield object


.. function:: starmap(function, iterable)

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

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


.. function:: takewhile(predicate, iterable)

   Make an iterator that returns elements from the iterable as long as the
   predicate is true.  Equivalent to::

      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


.. function:: tee(iterable, n=2)

   Return *n* independent iterators from a single iterable.  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)

   Once :func:`tee` has made a split, the original *iterable* should not be
   used anywhere else; otherwise, the *iterable* could get advanced without
   the tee objects being informed.

   This itertool may require significant auxiliary storage (depending on how
   much temporary data needs to be stored). In general, if one iterator uses
   most or all of the data before another iterator starts, it is faster to use
   :func:`list` instead of :func:`tee`.


.. function:: zip_longest(*iterables, fillvalue=None)

   Make an iterator that aggregates elements from each of the iterables. If the
   iterables are of uneven length, missing values are filled-in with *fillvalue*.
   Iteration continues until the longest iterable is exhausted.  Equivalent to::

      def zip_longest(*args, fillvalue=None):
          # zip_longest('ABCD', 'xy', fillvalue='-') --> Ax By C- D-
          def sentinel(counter = ([fillvalue]*(len(args)-1)).pop):
              yield counter()         # yields the fillvalue, or raises IndexError
          fillers = repeat(fillvalue)
          iters = [chain(it, sentinel(), fillers) for it in args]
          try:
              for tup in zip(*iters):
                  yield tup
          except IndexError:
              pass

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


.. _itertools-recipes:

Itertools Recipes
-----------------

This section shows recipes for creating an extended toolset using the existing
itertools as building blocks.

The extended tools offer the same high performance as the underlying toolset.
The superior memory performance is kept by processing elements one at a time
rather than bringing the whole iterable into memory all at once. Code volume is
kept small by linking the tools together in a functional style which helps
eliminate temporary variables.  High speed is retained by preferring
"vectorized" building blocks over the use of for-loops and :term:`generator`\s
which incur interpreter overhead.

.. testcode::

   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 map(function, count(start))

   def consume(iterator, n):
       "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 quantify(iterable, pred=bool):
       "Count how many times the predicate is true"
       return sum(map(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(map(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 zip(a, b)

   def grouper(n, iterable, fillvalue=None):
       "grouper(3, 'ABCDEFG', 'x') --> ABC DEF Gxx"
       args = [iter(iterable)] * n
       return zip_longest(*args, fillvalue=fillvalue)

   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 partition(pred, iterable):
       'Use a predicate to partition entries into false entries and true entries'
       # partition(is_odd, range(10)) --> 0 2 4 6 8   and  1 3 5 7 9
       t1, t2 = tee(iterable)
       return filterfalse(pred, t1), filter(pred, t2)

   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 filterfalse(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 map(next, map(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:
           iter_except(functools.partial(heappop, h), IndexError)   # priority queue iterator
           iter_except(d.popitem, KeyError)                         # non-blocking dict iterator
           iter_except(d.popleft, IndexError)                       # non-blocking deque iterator
           iter_except(q.get_nowait, Queue.Empty)                   # loop over a producer Queue
           iter_except(s.pop, KeyError)                             # non-blocking set iterator

       """
       try:
           if first is not None:
               yield first()            # For database APIs needing an initial cast to db.first()
           while 1:
               yield func()
       except exception:
           pass

   def random_product(*args, repeat=1):
       "Random selection from itertools.product(*args, **kwds)"
       pools = [tuple(pool) for pool in args] * repeat
       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(range(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 range(r))
       return tuple(pool[i] for i in indices)

Note, many of the above recipes can be optimized by replacing global lookups
with local variables defined as default values.  For example, the
*dotproduct* recipe can be written as::

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