"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), ...".  The same 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(starmap(operator.mul, zip(vec1, vec2,
strict=True)))".

**Infinite iterators:**

+--------------------+-------------------+---------------------------------------------------+-------------------------------------------+
| Iterator           | Arguments         | Results                                           | Example                                   |
|====================|===================|===================================================|===========================================|
| "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, … endlessly or up to n times    | "repeat(10, 3) --> 10 10 10"              |
+--------------------+-------------------+---------------------------------------------------+-------------------------------------------+

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

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

**Combinatoric iterators:**

+------------------------------------------------+----------------------+---------------------------------------------------------------+
| Iterator                                       | Arguments            | Results                                                       |
|================================================|======================|===============================================================|
| "product()"                                    | p, q, … [repeat=1]   | cartesian product, equivalent to a nested for-loop            |
+------------------------------------------------+----------------------+---------------------------------------------------------------+
| "permutations()"                               | p[, r]               | r-length tuples, all possible orderings, no repeated elements |
+------------------------------------------------+----------------------+---------------------------------------------------------------+
| "combinations()"                               | p, r                 | r-length tuples, in sorted order, no repeated elements        |
+------------------------------------------------+----------------------+---------------------------------------------------------------+
| "combinations_with_replacement()"              | p, r                 | r-length tuples, in sorted order, with repeated elements      |
+------------------------------------------------+----------------------+---------------------------------------------------------------+

+------------------------------------------------+---------------------------------------------------------------+
| Examples                                       | Results                                                       |
|================================================|===============================================================|
| "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"                               |
+------------------------------------------------+---------------------------------------------------------------+


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.accumulate(iterable[, func, *, initial=None])

   Make an iterator that returns accumulated sums, or accumulated
   results of other binary functions (specified via the optional
   *func* argument).

   If *func* is supplied, it should be a function of two arguments.
   Elements of the input *iterable* may be any type that can be
   accepted as arguments to *func*. (For example, with the default
   operation of addition, elements may be any addable type including
   "Decimal" or "Fraction".)

   Usually, the number of elements output matches the input iterable.
   However, if the keyword argument *initial* is provided, the
   accumulation leads off with the *initial* value so that the output
   has one more element than the input iterable.

   Roughly equivalent to:

      def accumulate(iterable, func=operator.add, *, initial=None):
          'Return running totals'
          # accumulate([1,2,3,4,5]) --> 1 3 6 10 15
          # accumulate([1,2,3,4,5], initial=100) --> 100 101 103 106 110 115
          # accumulate([1,2,3,4,5], operator.mul) --> 1 2 6 24 120
          it = iter(iterable)
          total = initial
          if initial is None:
              try:
                  total = next(it)
              except StopIteration:
                  return
          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 "min()" for a running minimum, "max()" for a running maximum, or
   "operator.mul()" for a running product.  Amortization tables can be
   built by accumulating interest and applying payments:

      >>> 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]

   See "functools.reduce()" for a similar function that returns only
   the final accumulated value.

   New in version 3.2.

   Changed in version 3.3: Added the optional *func* parameter.

   Changed in version 3.8: Added the optional *initial* parameter.

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. Roughly 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 "chain()".  Gets chained inputs from a
   single iterable argument that is evaluated lazily.  Roughly
   equivalent to:

      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*.

   The combination tuples are emitted in lexicographic ordering
   according to the order of the input *iterable*. So, if the input
   *iterable* is sorted, the output 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
   repeated values in each combination.

   Roughly 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.

   The combination tuples are emitted in lexicographic ordering
   according to the order of the input *iterable*. So, if the input
   *iterable* is sorted, the output 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.

   Roughly 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 has been exhausted. Roughly 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
   number *start*. Often used as an argument to "map()" to generate
   consecutive data points. Also, used with "zip()" to add sequence
   numbers.  Roughly 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())".

   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.  Roughly 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.
   Roughly 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. Roughly 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=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 "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 roughly 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):
              self.id = object()
              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, self.id))

          def _grouper(self, tgtkey, id):
              while self.id is id and self.currkey == tgtkey:
                  yield self.currvalue
                  try:
                      self.currvalue = next(self.it)
                  except StopIteration:
                      return
                  self.currkey = self.keyfunc(self.currvalue)

itertools.islice(iterable, stop)
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.

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

   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).

   Roughly 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)
          start, stop, step = s.start or 0, s.stop or sys.maxsize, s.step or 1
          it = iter(range(start, stop, step))
          try:
              nexti = next(it)
          except StopIteration:
              # Consume *iterable* up to the *start* position.
              for i, element in zip(range(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 zip(range(i + 1, stop), iterable):
                  pass

itertools.pairwise(iterable)

   Return successive overlapping pairs taken from the input
   *iterable*.

   The number of 2-tuples in the output iterator will be one fewer
   than the number of inputs.  It will be empty if the input iterable
   has fewer than two values.

   Roughly equivalent to:

      def pairwise(iterable):
          # pairwise('ABCDEFG') --> AB BC CD DE EF FG
          a, b = tee(iterable)
          next(b, None)
          return zip(a, b)

   New in version 3.10.

itertools.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.

   The permutation tuples are emitted in lexicographic order according
   to the order of the input *iterable*. So, if the input *iterable*
   is sorted, the output 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
   repeated values within a permutation.

   Roughly 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 = list(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=1)

   Cartesian product of input iterables.

   Roughly 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 roughly 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)

   Before "product()" runs, it completely consumes the input
   iterables, keeping pools of values in memory to generate the
   products.  Accordingly, it is only useful with finite inputs.

itertools.repeat(object[, times])

   Make an iterator that returns *object* over and over again. Runs
   indefinitely unless the *times* argument is specified.

   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 range(times):
                  yield object

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

      >>> list(map(pow, range(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 "map()" when argument
   parameters are already grouped in tuples from a single iterable
   (when the data has been “pre-zipped”).

   The difference between "map()" 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)

itertools.takewhile(predicate, iterable)

   Make an iterator that returns elements from the iterable as long as
   the predicate is true.  Roughly 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.

   The following Python code helps explain what *tee* does (although
   the actual implementation is more complex and uses only a single
   underlying FIFO (first-in, first-out) queue):

      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
                      try:
                          newval = next(it)   # fetch a new value and
                      except StopIteration:
                          return
                      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 a "tee()" has been created, the original *iterable* should not
   be used anywhere else; otherwise, the *iterable* could get advanced
   without the tee objects being informed.

   "tee" iterators are not threadsafe. A "RuntimeError" may be raised
   when using simultaneously iterators returned by the same "tee()"
   call, even if the original *iterable* is threadsafe.

   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=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.  Roughly equivalent to:

      def zip_longest(*args, fillvalue=None):
          # zip_longest('ABCD', 'xy', fillvalue='-') --> Ax By C- D-
          iterators = [iter(it) for it in args]
          num_active = len(iterators)
          if not num_active:
              return
          while True:
              values = []
              for i, it in enumerate(iterators):
                  try:
                      value = next(it)
                  except StopIteration:
                      num_active -= 1
                      if not num_active:
                          return
                      iterators[i] = repeat(fillvalue)
                      value = fillvalue
                  values.append(value)
              yield tuple(values)

   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".


Itertools Recipes
=================

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

The primary purpose of the itertools recipes is educational.  The
recipes show various ways of thinking about individual tools — for
example, that "chain.from_iterable" is related to the concept of
flattening.  The recipes also give ideas about ways that the tools can
be combined — for example, how "compress()" and "range()" can work
together.  The recipes also show patterns for using itertools with the
"operator" and "collections" modules as well as with the built-in
itertools such as "map()", "filter()", "reversed()", and
"enumerate()".

A secondary purpose of the recipes is to serve as an incubator.  The
"accumulate()", "compress()", and "pairwise()" itertools started out
as recipes.  Currently, the "iter_index()" recipe is being tested to
see whether it proves its worth.

Substantially all of these recipes and many, many others can be
installed from the more-itertools project found on the Python Package
Index:

   python -m pip install more-itertools

Many of the recipes offer the same high performance as the underlying
toolset. 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.

   import collections
   import math
   import operator
   import random

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

   def prepend(value, iterable):
       "Prepend a single value in front of an iterable"
       # prepend(1, [2, 3, 4]) --> 1 2 3 4
       return chain([value], iterable)

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

   def tail(n, iterable):
       "Return an iterator over the last n items"
       # tail(3, 'ABCDEFG') --> E F G
       return iter(collections.deque(iterable, maxlen=n))

   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(map(pred, iterable))

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

   def batched(iterable, n):
       "Batch data into tuples of length n. The last batch may be shorter."
       # batched('ABCDEFG', 3) --> ABC DEF G
       if n < 1:
           raise ValueError('n must be at least one')
       it = iter(iterable)
       while batch := tuple(islice(it, n)):
           yield batch

   def grouper(iterable, n, *, incomplete='fill', fillvalue=None):
       "Collect data into non-overlapping fixed-length chunks or blocks"
       # grouper('ABCDEFG', 3, fillvalue='x') --> ABC DEF Gxx
       # grouper('ABCDEFG', 3, incomplete='strict') --> ABC DEF ValueError
       # grouper('ABCDEFG', 3, incomplete='ignore') --> ABC DEF
       args = [iter(iterable)] * n
       if incomplete == 'fill':
           return zip_longest(*args, fillvalue=fillvalue)
       if incomplete == 'strict':
           return zip(*args, strict=True)
       if incomplete == 'ignore':
           return zip(*args)
       else:
           raise ValueError('Expected fill, strict, or ignore')

   def sumprod(vec1, vec2):
       "Compute a sum of products."
       return sum(starmap(operator.mul, zip(vec1, vec2, strict=True)))

   def sum_of_squares(it):
       "Add up the squares of the input values."
       # sum_of_squares([10, 20, 30]) -> 1400
       return sumprod(*tee(it))

   def transpose(it):
       "Swap the rows and columns of the input."
       # transpose([(1, 2, 3), (11, 22, 33)]) --> (1, 11) (2, 22) (3, 33)
       return zip(*it, strict=True)

   def matmul(m1, m2):
       "Multiply two matrices."
       # matmul([(7, 5), (3, 5)], [[2, 5], [7, 9]]) --> (49, 80), (41, 60)
       n = len(m2[0])
       return batched(starmap(sumprod, product(m1, transpose(m2))), n)

   def convolve(signal, kernel):
       # See:  https://betterexplained.com/articles/intuitive-convolution/
       # convolve(data, [0.25, 0.25, 0.25, 0.25]) --> Moving average (blur)
       # convolve(data, [1, -1]) --> 1st finite difference (1st derivative)
       # convolve(data, [1, -2, 1]) --> 2nd finite difference (2nd derivative)
       kernel = tuple(kernel)[::-1]
       n = len(kernel)
       window = collections.deque([0], maxlen=n) * n
       for x in chain(signal, repeat(0, n-1)):
           window.append(x)
           yield sumprod(kernel, window)

   def polynomial_from_roots(roots):
       """Compute a polynomial's coefficients from its roots.

          (x - 5) (x + 4) (x - 3)  expands to:   x³ -4x² -17x + 60
       """
       # polynomial_from_roots([5, -4, 3]) --> [1, -4, -17, 60]
       expansion = [1]
       for r in roots:
           expansion = convolve(expansion, (1, -r))
       return list(expansion)

   def polynomial_eval(coefficients, x):
       """Evaluate a polynomial at a specific value.

       Computes with better numeric stability than Horner's method.
       """
       # Evaluate x³ -4x² -17x + 60 at x = 2.5
       # polynomial_eval([1, -4, -17, 60], x=2.5) --> 8.125
       n = len(coefficients)
       if n == 0:
           return x * 0  # coerce zero to the type of x
       powers = map(pow, repeat(x), reversed(range(n)))
       return sumprod(coefficients, powers)

   def iter_index(iterable, value, start=0):
       "Return indices where a value occurs in a sequence or iterable."
       # iter_index('AABCADEAF', 'A') --> 0 1 4 7
       try:
           seq_index = iterable.index
       except AttributeError:
           # Slow path for general iterables
           it = islice(iterable, start, None)
           i = start - 1
           try:
               while True:
                   yield (i := i + operator.indexOf(it, value) + 1)
           except ValueError:
               pass
       else:
           # Fast path for sequences
           i = start - 1
           try:
               while True:
                   yield (i := seq_index(value, i+1))
           except ValueError:
               pass

   def sieve(n):
       "Primes less than n"
       # sieve(30) --> 2 3 5 7 11 13 17 19 23 29
       data = bytearray((0, 1)) * (n // 2)
       data[:3] = 0, 0, 0
       limit = math.isqrt(n) + 1
       for p in compress(range(limit), data):
           data[p*p : n : p+p] = bytes(len(range(p*p, n, p+p)))
       data[2] = 1
       return iter_index(data, 1) if n > 2 else iter([])

   def factor(n):
       "Prime factors of n."
       # factor(99) --> 3 3 11
       for prime in sieve(math.isqrt(n) + 1):
           while True:
               quotient, remainder = divmod(n, prime)
               if remainder:
                   break
               yield prime
               n = quotient
               if n == 1:
                   return
       if n > 1:
           yield n

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

   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 triplewise(iterable):
       "Return overlapping triplets from an iterable"
       # triplewise('ABCDEFG') --> ABC BCD CDE DEF EFG
       for (a, _), (b, c) in pairwise(pairwise(iterable)):
           yield a, b, c

   def sliding_window(iterable, n):
       # sliding_window('ABCDEFG', 4) --> ABCD BCDE CDEF DEFG
       it = iter(iterable)
       window = collections.deque(islice(it, n), maxlen=n)
       if len(window) == n:
           yield tuple(window)
       for x in it:
           window.append(x)
           yield tuple(window)

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

   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 before_and_after(predicate, it):
       """ Variant of takewhile() that allows complete
           access to the remainder of the iterator.

           >>> it = iter('ABCdEfGhI')
           >>> all_upper, remainder = before_and_after(str.isupper, it)
           >>> ''.join(all_upper)
           'ABC'
           >>> ''.join(remainder)     # takewhile() would lose the 'd'
           'dEfGhI'

           Note that the first iterator must be fully
           consumed before the second iterator can
           generate valid results.
       """
       it = iter(it)
       transition = []
       def true_iterator():
           for elem in it:
               if predicate(elem):
                   yield elem
               else:
                   transition.append(elem)
                   return
       def remainder_iterator():
           yield from transition
           yield from it
       return true_iterator(), remainder_iterator()

   def subslices(seq):
       "Return all contiguous non-empty subslices of a sequence"
       # subslices('ABCD') --> A AB ABC ABCD B BC BCD C CD D
       slices = starmap(slice, combinations(range(len(seq) + 1), 2))
       return map(operator.getitem, repeat(seq), slices)

   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()
       if key is None:
           for element in filterfalse(seen.__contains__, iterable):
               seen.add(element)
               yield element
           # For order preserving deduplication,
           # a faster but non-lazy solution is:
           #     yield from dict.fromkeys(iterable)
       else:
           for element in iterable:
               k = key(element)
               if k not in seen:
                   seen.add(k)
                   yield element
           # For use cases that allow the last matching element to be returned,
           # a faster but non-lazy solution is:
           #      t1, t2 = tee(iterable)
           #      yield from dict(zip(map(key, t1), t2)).values()

   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(operator.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 builtins.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 True:
               yield func()
       except exception:
           pass

   def first_true(iterable, default=False, pred=None):
       """Returns the first true value in the iterable.

       If no true value is found, returns *default*

       If *pred* is not None, returns the first item
       for which pred(item) is true.

       """
       # first_true([a,b,c], x) --> a or b or c or x
       # first_true([a,b], x, f) --> a if f(a) else b if f(b) else x
       return next(filter(pred, iterable), default)

   def nth_combination(iterable, r, index):
       "Equivalent to list(combinations(iterable, r))[index]"
       pool = tuple(iterable)
       n = len(pool)
       c = math.comb(n, r)
       if index < 0:
           index += c
       if index < 0 or index >= c:
           raise IndexError
       result = []
       while r:
           c, n, r = c*r//n, n-1, r-1
           while index >= c:
               index -= c
               c, n = c*(n-r)//n, n-1
           result.append(pool[-1-n])
       return tuple(result)
