
``itertools`` --- Functions creating iterators for efficient looping
********************************************************************

This module implements a number of *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), ...``.  But, this effect can be
achieved in Python by combining ``map()`` and ``count()`` to form
``map(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(map(operator.mul, vector1, vector2))``.

**Infinite Iterators:**

   +--------------------+-------------------+---------------------------------------------------+
   | Iterator           | Arguments         | Results                                           |
   +====================+===================+===================================================+
   | ``count()``        | start, [step]     | start, start+step, start+2*step, ...              |
   +--------------------+-------------------+---------------------------------------------------+
   | ``cycle()``        | p                 | p0, p1, ... plast, p0, p1, ...                    |
   +--------------------+-------------------+---------------------------------------------------+
   | ``repeat()``       | elem [,n]         | elem, elem, elem, ... endlessly or up to n times  |
   +--------------------+-------------------+---------------------------------------------------+

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

   +----------------------+------------------------------+---------------------------------------------------+
   | Iterator             | Arguments                    | Results                                           |
   +======================+==============================+===================================================+
   | ``chain()``          | p, q, ...                    | p0, p1, ... plast, q0, q1, ...                    |
   +----------------------+------------------------------+---------------------------------------------------+
   | ``compress()``       | data, selectors              | (d[0] if s[0]), (d[1] if s[1]), ...               |
   +----------------------+------------------------------+---------------------------------------------------+
   | ``dropwhile()``      | pred, seq                    | seq[n], seq[n+1], starting when pred fails        |
   +----------------------+------------------------------+---------------------------------------------------+
   | ``filterfalse()``    | pred, seq                    | elements of seq where pred(elem) is False         |
   +----------------------+------------------------------+---------------------------------------------------+
   | ``groupby()``        | iterable[, keyfunc]          | sub-iterators grouped by value of keyfunc(v)      |
   +----------------------+------------------------------+---------------------------------------------------+
   | ``islice()``         | seq, [start,] stop [, step]  | elements from seq[start:stop:step]                |
   +----------------------+------------------------------+---------------------------------------------------+
   | ``starmap()``        | func, seq                    | func(*seq[0]), func(*seq[1]), ...                 |
   +----------------------+------------------------------+---------------------------------------------------+
   | ``tee()``            | it, n                        | it1, it2 , ... itn  splits one iterator into n    |
   +----------------------+------------------------------+---------------------------------------------------+
   | ``takewhile()``      | pred, seq                    | seq[0], seq[1], until pred fails                  |
   +----------------------+------------------------------+---------------------------------------------------+
   | ``zip_longest()``    | p, q, ...                    | (p[0], q[0]), (p[1], q[1]), ...                   |
   +----------------------+------------------------------+---------------------------------------------------+

**Combinatoric generators:**

   +---------------------------------------+----------------------+---------------------------------------------------+
   | Iterator                              | Arguments            | Results                                           |
   +=======================================+======================+===================================================+
   | ``product()``                         | p, q, ... [repeat=1] | cartesian product                                 |
   +---------------------------------------+----------------------+---------------------------------------------------+
   | ``permutations()``                    | p[, r]               | r-length permutations (without repeated elements) |
   +---------------------------------------+----------------------+---------------------------------------------------+
   | ``combinations()``                    | p[, r]               | r-length combinations (sorted and no repeats)     |
   +---------------------------------------+----------------------+---------------------------------------------------+
   | ``combinations_with_replacement()``   | p[, r]               | r-length combinations (sorted but with repeats)   |
   +---------------------------------------+----------------------+---------------------------------------------------+


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.

itertools.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

itertools.chain.from_iterable(iterable)

   Alternate constructor for ``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

itertools.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 ``combinations()`` can be also expressed as a
   subsequence of ``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``.

itertools.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 ``combinations_with_replacement()`` can be also
   expressed as a subsequence of ``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``.

   New in version 3.1.

itertools.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 have 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)

   New in version 3.1.

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

   Make an iterator that returns evenly spaced values starting with
   *n*. Often used as an argument to ``map()`` to generate consecutive
   data points. Also, used with ``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) -> 3.5 3.0 4.5 ...
          n = start
          while True:
              yield n
              n += step

   Changed in version 3.1: added *step* argument and allowed non-
   integer arguments.

itertools.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).

itertools.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

itertools.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

itertools.groupby(iterable[, key])

   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 ``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 ``groupby()``.  Because the source is shared, when
   the ``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)

   ``groupby()`` is equivalent to:

      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)

itertools.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,
   ``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 = 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.

itertools.permutations(iterable[, r])

   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 ``permutations()`` can be also expressed as a
   subsequence of ``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``.

itertools.product(*iterables[, repeat])

   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 = map(tuple, 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)

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 ``map()`` for invariant parameters to the called
   function.  Also used with ``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

itertools.starmap(function, iterable)

   Make an iterator that computes the function using arguments
   obtained from the iterable.  Used instead of ``map()`` when
   argument parameters are already grouped in tuples from a single
   iterable (the data has been "pre-zipped").  The difference between
   ``map()`` and ``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)

itertools.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

itertools.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 ``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 ``list()`` instead of ``tee()``.

itertools.zip_longest(*iterables[, fillvalue])

   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
   ``zip_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``.


Examples
========

The following examples show common uses for each tool and demonstrate
ways they can be combined.

   >>> # Show a dictionary sorted and grouped by value
   >>> from operator import itemgetter
   >>> d = dict(a=1, b=2, c=1, d=2, e=1, f=2, g=3)
   >>> di = sorted(d.items(), key=itemgetter(1))
   >>> for k, g in groupby(di, key=itemgetter(1)):
   ...     print(k, map(itemgetter(0), g))
   ...
   1 ['a', 'c', 'e']
   2 ['b', 'd', 'f']
   3 ['g']

   >>> # Find runs of consecutive numbers using groupby.  The key to the solution
   >>> # is differencing with a range so that consecutive numbers all appear in
   >>> # same group.
   >>> data = [ 1,  4,5,6, 10, 15,16,17,18, 22, 25,26,27,28]
   >>> for k, g in groupby(enumerate(data), lambda t:t[0]-t[1]):
   ...     print(map(operator.itemgetter(1), g))
   ...
   [1]
   [4, 5, 6]
   [10]
   [15, 16, 17, 18]
   [22]
   [25, 26, 27, 28]


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 *generator*s which incur
interpreter overhead.

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

   def enumerate(iterable, start=0):
       return zip(count(start), iterable)

   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."
       collections.deque(islice(iterator, n), maxlen=0)

   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(iterable, n))

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

   def flatten(listOfLists):
       return list(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 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 iterable:
               if element not in seen:
                   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)))
