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# 9.1. `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(map(operator.mul, vector1, vector2))`.

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 | 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`
`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[, keyfunc] | sub-iterators grouped by value of keyfunc(v) |
`islice()` | seq, [start,] stop [, step] | elements from seq[start:stop:step] | `islice('ABCDEFG', 2, None) --> C D E F G`
`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 generators:

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
`product('ABCD', repeat=2)` |  | `AA AB AC AD BA BB BC BD CA CB CC CD DA DB DC DD`
`permutations('ABCD', 2)` |  | `AB AC AD BA BC BD CA CB CD DA DB DC`
`combinations('ABCD', 2)` |  | `AB AC AD BC BD CD`
`combinations_with_replacement('ABCD', 2)` |  | `AA AB AC AD BB BC BD CC CD DD`

## 9.1.1. 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)

  Make an iterator that returns accumulated sums. Elements may be any addable type including `Decimal` or `Fraction`. Equivalent to:

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

  New in version 3.2.

`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

classmethod `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 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)

  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) -> 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. 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=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 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)

`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 = 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.

`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.

  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 = 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.

  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)

`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

  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 (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=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:

      class ZipExhausted(Exception):
          pass

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

  If one of the iterables is potentially infinite, then the `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`.

## 9.1.2. 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 generators which incur interpreter overhead.

    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))

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