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

.. module:: itertools :synopsis: Functions creating iterators for efficient looping.

.. testsetup::

from itertools import * import collections import math import operator import random

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

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

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

General iterators:

============================ Iterator ============================ :func:accumulate :func:batched :func:chain :func:chain.from_iterable :func:compress :func:count :func:cycle :func:dropwhile :func:filterfalse :func:groupby :func:islice :func:pairwise :func:repeat :func:starmap :func:takewhile :func:tee :func:zip_longest ============================ ============================ ================================================= ============================================================= Arguments Results Example ============================ ================================================= ============================================================= p [,func] p0, p0+p1, p0+p1+p2, ... accumulate([1,2,3,4,5]) → 1 3 6 10 15 p, n (p0, p1, ..., p_n-1), ... batched('ABCDEFG', n=3) → ABC DEF G p, q, ... p0, p1, ... plast, q0, q1, ... chain('ABC', 'DEF') → A B C D E F iterable p0, p1, ... plast, q0, q1, ... chain.from_iterable(['ABC', 'DEF']) → A B C D E F data, selectors (d[0] if s[0]), (d[1] if s[1]), ... compress('ABCDEF', [1,0,1,0,1,1]) → A C E F [start[, step]] start, start+step, start+2*step, ... count(10) → 10 11 12 13 14 ... p p0, p1, ... plast, p0, p1, ... cycle('ABCD') → A B C D A B C D ... predicate, seq seq[n], seq[n+1], starting when predicate fails dropwhile(lambda x: x<5, [1,4,6,3,8]) → 6 3 8 predicate, seq elements of seq where predicate(elem) fails filterfalse(lambda x: x<5, [1,4,6,3,8]) → 6 8 iterable[, key] sub-iterators grouped by value of key(v) groupby(['A','B','DEF'], len) → (1, A B) (3, DEF) seq, [start,] stop [, step] elements from seq[start:stop:step] islice('ABCDEFG', 2, None) → C D E F G iterable (p[0], p[1]), (p[1], p[2]) pairwise('ABCDEFG') → AB BC CD DE EF FG elem [,n] elem, elem, elem, ... endlessly or up to n times repeat(10, 3) → 10 10 10 func, seq func(*seq[0]), func(*seq[1]), ... starmap(pow, [(2,5), (3,2), (10,3)]) → 32 9 1000 predicate, seq seq[0], seq[1], until predicate fails takewhile(lambda x: x<5, [1,4,6,3,8]) → 1 4 it, n it1, it2, ... itn splits one iterator into n tee('ABC', 2) → A B C, A B C p, q, ... (p[0], q[0]), (p[1], q[1]), ... zip_longest('ABCD', 'xy', fillvalue='-') → Ax By C- D- ============================ ================================================= =============================================================

Combinatoric iterators:

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

============================================== ============================================================= 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 ============================================== =============================================================

.. _itertools-functions:

Itertool Functions

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

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

Make an iterator that returns accumulated sums or accumulated
results from other binary functions.

The *function* defaults to addition.  The *function* should accept
two arguments, an accumulated total and a value from the *iterable*.

If an *initial* value is provided, the accumulation will start with
that value and the output will have one more element than the input
iterable.

Roughly equivalent to::

    def accumulate(iterable, function=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

        iterator = iter(iterable)
        total = initial
        if initial is None:
            try:
                total = next(iterator)
            except StopIteration:
                return

        yield total
        for element in iterator:
            total = function(total, element)
            yield total

To compute a running minimum, set *function* to :func:`min`.
For a running maximum, set *function* to :func:`max`.
Or for a running product, set *function* to :func:`operator.mul`.
To build an `amortization table
<https://www.ramseysolutions.com/real-estate/amortization-schedule>`_,
accumulate the interest and apply payments:

.. doctest::

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

  # Amortize a 5% loan of 1000 with 10 annual payments of 90
  >>> update = lambda balance, payment: round(balance * 1.05) - payment
  >>> list(accumulate(repeat(90, 10), update, initial=1_000))
  [1000, 960, 918, 874, 828, 779, 728, 674, 618, 559, 497]

See :func:`functools.reduce` for a similar function that returns only the
final accumulated value.

.. versionadded:: 3.2

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

.. versionchanged:: 3.8
   Added the optional *initial* parameter.

.. function:: batched(iterable, n, *, strict=False)

Batch data from the iterable into tuples of length n. The last batch may be shorter than n.

If strict is true, will raise a :exc:ValueError if the final batch is shorter than n.

Loops over the input iterable and accumulates data into tuples up to size n. The input is consumed lazily, just enough to fill a batch. The result is yielded as soon as the batch is full or when the input iterable is exhausted:

.. doctest::

  >>> flattened_data = ['roses', 'red', 'violets', 'blue', 'sugar', 'sweet']
  >>> unflattened = list(batched(flattened_data, 2))
  >>> unflattened
  [('roses', 'red'), ('violets', 'blue'), ('sugar', 'sweet')]

Roughly equivalent to::

  def batched(iterable, n, *, strict=False):
      # batched('ABCDEFG', 3) → ABC DEF G
      if n < 1:
          raise ValueError('n must be at least one')
      iterator = iter(iterable)
      while batch := tuple(islice(iterator, n)):
          if strict and len(batch) != n:
              raise ValueError('batched(): incomplete batch')
          yield batch

.. versionadded:: 3.12

.. versionchanged:: 3.13 Added the strict option.

.. function:: chain(*iterables)

Make an iterator that returns elements from the first iterable until it is exhausted, then proceeds to the next iterable, until all of the iterables are exhausted. This combines multiple data sources into a single iterator. Roughly equivalent to::

  def chain(*iterables):
      # chain('ABC', 'DEF') → A B C D E F
      for iterable in iterables:
          yield from iterable

.. classmethod:: chain.from_iterable(iterable)

Alternate constructor for :func: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 iterable in iterables:
          yield from iterable

.. function:: combinations(iterable, r)

Return r length subsequences of elements from the input iterable.

The output is a subsequence of :func:product keeping only entries that are subsequences of the iterable. The length of the output is given by :func:math.comb which computes n! / r! / (n - r)! when 0 ≤ r ≤ n or zero when r > n.

The combination tuples are emitted in lexicographic order according to the order of the input iterable. 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. If the input elements are unique, there will be no repeated values within 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)

.. function:: combinations_with_replacement(iterable, r)

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

The output is a subsequence of :func:product that keeps only entries that are subsequences (with possible repeated elements) of the iterable. The number of subsequence returned is (n + r - 1)! / r! / (n - 1)! when n > 0.

The combination tuples are emitted in lexicographic order according to the order of the input iterable. 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. 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)

.. versionadded:: 3.1

.. function:: compress(data, selectors)

Make an iterator that returns elements from data where the corresponding element in selectors is true. Stops when either the data or selectors iterables have been exhausted. Roughly equivalent to::

   def compress(data, selectors):
       # compress('ABCDEF', [1,0,1,0,1,1]) → A C E F
       return (datum for datum, selector in zip(data, selectors) if selector)

.. versionadded:: 3.1

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

Make an iterator that returns evenly spaced values beginning with start. Can be used with :func:map to generate consecutive data points or with :func: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()).

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

.. function:: cycle(iterable)

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

This itertool may require significant auxiliary storage (depending on the length of the iterable).

.. function:: dropwhile(predicate, iterable)

Make an iterator that drops elements from the iterable while the predicate is true and afterwards returns every element. Roughly equivalent to::

  def dropwhile(predicate, iterable):
      # dropwhile(lambda x: x<5, [1,4,6,3,8]) → 6 3 8

      iterator = iter(iterable)
      for x in iterator:
          if not predicate(x):
              yield x
              break

      for x in iterator:
          yield x

Note this does not produce any output until the predicate first becomes false, so this itertool may have a lengthy start-up time.

.. function:: filterfalse(predicate, iterable)

Make an iterator that filters elements from the iterable returning only those for which the predicate returns a false value. If predicate is None, returns the items that are false. Roughly equivalent to::

  def filterfalse(predicate, iterable):
      # filterfalse(lambda x: x<5, [1,4,6,3,8]) → 6 8

      if predicate is None:
          predicate = bool

      for x in iterable:
          if not predicate(x):
              yield x

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

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

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

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

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

:func:groupby is roughly equivalent to::

  def groupby(iterable, key=None):
      # [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

      keyfunc = (lambda x: x) if key is None else key
      iterator = iter(iterable)
      exhausted = False

      def _grouper(target_key):
          nonlocal curr_value, curr_key, exhausted
          yield curr_value
          for curr_value in iterator:
              curr_key = keyfunc(curr_value)
              if curr_key != target_key:
                  return
              yield curr_value
          exhausted = True

      try:
          curr_value = next(iterator)
      except StopIteration:
          return
      curr_key = keyfunc(curr_value)

      while not exhausted:
          target_key = curr_key
          curr_group = _grouper(target_key)
          yield curr_key, curr_group
          if curr_key == target_key:
              for _ in curr_group:
                  pass

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

Make an iterator that returns selected elements from the iterable. Works like sequence slicing but does not support negative values for start, stop, or step.

If start is zero or None, iteration starts at zero. Otherwise, elements from the iterable are skipped until start is reached.

If stop is None, iteration continues until the input is exhausted, if at all. Otherwise, it stops at the specified position.

If step is None, the step defaults to one. Elements are returned consecutively unless step is set higher than one which results in items being skipped.

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 = 0 if s.start is None else s.start
      stop = s.stop
      step = 1 if s.step is None else s.step
      if start < 0 or (stop is not None and stop < 0) or step <= 0:
          raise ValueError

      indices = count() if stop is None else range(max(start, stop))
      next_i = start
      for i, element in zip(indices, iterable):
          if i == next_i:
              yield element
              next_i += step

If the input is an iterator, then fully consuming the islice advances the input iterator by max(start, stop) steps regardless of the step value.

.. function:: 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

        iterator = iter(iterable)
        a = next(iterator, None)

        for b in iterator:
            yield a, b
            a = b

.. versionadded:: 3.10

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

Return successive r length permutations of elements <https://www.britannica.com/science/permutation>_ from 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 output is a subsequence of :func:product where entries with repeated elements have been filtered out. The length of the output is given by :func:math.perm which computes n! / (n - r)! when 0 ≤ r ≤ n or zero when r > n.

The permutation tuples are emitted in lexicographic order according to the order of the input iterable. 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. 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

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

Cartesian product <https://en.wikipedia.org/wiki/Cartesian_product>_ of the 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(*iterables, 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

       if repeat < 0:
           raise ValueError('repeat argument cannot be negative')
       pools = [tuple(pool) for pool in iterables] * 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 :func: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.

.. function:: 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:

.. doctest::

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

.. function:: starmap(function, iterable)

Make an iterator that computes the function using arguments obtained from the iterable. Used instead of :func:map when argument parameters have already been "pre-zipped" into tuples.

The difference between :func:map and :func: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)

.. function:: 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, [1,4,6,3,8]) → 1 4
      for x in iterable:
          if not predicate(x):
              break
          yield x

Note, the element that first fails the predicate condition is consumed from the input iterator and there is no way to access it. This could be an issue if an application wants to further consume the input iterator after takewhile has been run to exhaustion. To work around this problem, consider using more-itertools before_and_after() <https://more-itertools.readthedocs.io/en/stable/api.html#more_itertools.before_and_after>_ instead.

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

Return n independent iterators from a single iterable.

Roughly equivalent to::

    def tee(iterable, n=2):
        if n < 0:
            raise ValueError
        if n == 0:
            return ()
        iterator = _tee(iterable)
        result = [iterator]
        for _ in range(n - 1):
            result.append(_tee(iterator))
        return tuple(result)

    class _tee:

        def __init__(self, iterable):
            it = iter(iterable)
            if isinstance(it, _tee):
                self.iterator = it.iterator
                self.link = it.link
            else:
                self.iterator = it
                self.link = [None, None]

        def __iter__(self):
            return self

        def __next__(self):
            link = self.link
            if link[1] is None:
                link[0] = next(self.iterator)
                link[1] = [None, None]
            value, self.link = link
            return value

When the input iterable is already a tee iterator object, all members of the return tuple are constructed as if they had been produced by the upstream :func:tee call. This "flattening step" allows nested :func:tee calls to share the same underlying data chain and to have a single update step rather than a chain of calls.

The flattening property makes tee iterators efficiently peekable:

.. testcode::

  def lookahead(tee_iterator):
       "Return the next value without moving the input forward"
       [forked_iterator] = tee(tee_iterator, 1)
       return next(forked_iterator)

.. doctest::

  >>> iterator = iter('abcdef')
  >>> [iterator] = tee(iterator, 1)   # Make the input peekable
  >>> next(iterator)                  # Move the iterator forward
  'a'
  >>> lookahead(iterator)             # Check next value
  'b'
  >>> next(iterator)                  # Continue moving forward
  'b'

tee iterators are not threadsafe. A :exc:RuntimeError may be raised when simultaneously using iterators returned by the same :func: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 :func:list instead of :func:tee.

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

Make an iterator that aggregates elements from each of the iterables.

If the iterables are of uneven length, missing values are filled-in with fillvalue. If not specified, fillvalue defaults to None.

Iteration continues until the longest iterable is exhausted.

Roughly equivalent to::

  def zip_longest(*iterables, fillvalue=None):
      # zip_longest('ABCD', 'xy', fillvalue='-') → Ax By C- D-

      iterators = list(map(iter, iterables))
      num_active = len(iterators)
      if not num_active:
          return

      while True:
          values = []
          for i, iterator in enumerate(iterators):
              try:
                  value = next(iterator)
              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 :func:zip_longest function should be wrapped with something that limits the number of calls (for example :func:islice or :func:takewhile).

.. _itertools-recipes:

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 starmap() and repeat() can work together. The recipes also show patterns for using itertools with the :mod:operator and :mod: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 sliding_window(), derangements(), and sieve() recipes are being tested to see whether they prove their worth.

Substantially all of these recipes and many, many others can be installed from the :pypi: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 <https://www.cs.kent.ac.uk/people/staff/dat/miranda/whyfp90.pdf>_. High speed is retained by preferring "vectorized" building blocks over the use of for-loops and :term:generators <generator> which incur interpreter overhead.

.. testcode::

from itertools import (accumulate, batched, chain, combinations, compress, count, cycle, filterfalse, groupby, islice, permutations, product, repeat, starmap, tee, zip_longest) from collections import Counter, deque from contextlib import suppress from functools import reduce from math import comb, isqrt, prod, sumprod from operator import getitem, is_not, itemgetter, mul, neg, truediv

==== Basic one liners ====

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 running_mean(iterable): "Yield the average of all values seen so far." # running_mean([8.5, 9.5, 7.5, 6.5]) -> 8.5 9.0 8.5 8.0 return map(truediv, accumulate(iterable), count(1))

def repeatfunc(function, times=None, *args): "Repeat calls to a function with specified arguments." if times is None: return starmap(function, repeat(args)) return starmap(function, repeat(args, times))

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

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

def loops(n): "Loop n times. Like range(n) but without creating integers." # for _ in loops(100): ... return repeat(None, n)

def tail(n, iterable): "Return an iterator over the last n items." # tail(3, 'ABCDEFG') → E F G return iter(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: deque(iterator, maxlen=0) else: 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, predicate=bool): "Given a predicate that returns True or False, count the True results." return sum(map(predicate, iterable))

def first_true(iterable, default=False, predicate=None): "Returns the first true value or the default if there is no true value." # 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(predicate, iterable), default)

def all_equal(iterable, key=None): "Returns True if all the elements are equal to each other." # all_equal('4٤௪౪໔', key=int) → True return len(take(2, groupby(iterable, key))) <= 1

==== Data pipelines ====

def unique_justseen(iterable, key=None): "Yield unique elements, preserving order. Remember only the element just seen." # unique_justseen('AAAABBBCCDAABBB') → A B C D A B # unique_justseen('ABBcCAD', str.casefold) → A B c A D if key is None: return map(itemgetter(0), groupby(iterable)) return map(next, map(itemgetter(1), groupby(iterable, key)))

def unique_everseen(iterable, key=None): "Yield unique elements, preserving order. Remember all elements ever seen." # unique_everseen('AAAABBBCCDAABBB') → A B C D # unique_everseen('ABBcCAD', str.casefold) → A B c D seen = set() 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(iterable, key=None, reverse=False): "Yield unique elements in sorted order. Supports unhashable inputs." # unique([[1, 2], [3, 4], [1, 2]]) → [1, 2] [3, 4] sequenced = sorted(iterable, key=key, reverse=reverse) return unique_justseen(sequenced, key=key)

def sliding_window(iterable, n): "Collect data into overlapping fixed-length chunks or blocks." # sliding_window('ABCDEFG', 3) → ABC BCD CDE DEF EFG iterator = iter(iterable) window = deque(islice(iterator, n - 1), maxlen=n) for x in iterator: window.append(x) yield tuple(window)

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 iterators = [iter(iterable)] * n match incomplete: case 'fill': return zip_longest(*iterators, fillvalue=fillvalue) case 'strict': return zip(*iterators, strict=True) case 'ignore': return zip(*iterators) case _: raise ValueError('Expected fill, strict, or ignore')

def roundrobin(*iterables): "Visit input iterables in a cycle until each is exhausted." # roundrobin('ABC', 'D', 'EF') → A D E B F C # Algorithm credited to George Sakkis iterators = map(iter, iterables) for num_active in range(len(iterables), 0, -1): iterators = cycle(islice(iterators, num_active)) yield from map(next, iterators)

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(getitem, repeat(seq), slices)

def derangements(iterable, r=None): "Produce r length permutations without fixed points." # derangements('ABCD') → BADC BCDA BDAC CADB CDAB CDBA DABC DCAB DCBA # Algorithm credited to Stefan Pochmann seq = tuple(iterable) pos = tuple(range(len(seq))) have_moved = map(map, repeat(is_not), repeat(pos), permutations(pos, r=r)) valid_derangements = map(all, have_moved) return compress(permutations(seq, r=r), valid_derangements)

def iter_index(iterable, value, start=0, stop=None): "Return indices where a value occurs in a sequence or iterable." # iter_index('AABCADEAF', 'A') → 0 1 4 7 seq_index = getattr(iterable, 'index', None) if seq_index is None: iterator = islice(iterable, start, stop) for i, element in enumerate(iterator, start): if element is value or element == value: yield i else: stop = len(iterable) if stop is None else stop i = start with suppress(ValueError): while True: yield (i := seq_index(value, i, stop)) i += 1

def iter_except(function, exception, first=None): "Convert a call-until-exception interface to an iterator interface." # iter_except(d.popitem, KeyError) → non-blocking dictionary iterator with suppress(exception): if first is not None: yield first() while True: yield function()

==== Mathematical operations ====

def multinomial(*counts): "Number of distinct arrangements of a multiset." # Counter('abracadabra').values() → 5 2 2 1 1 # multinomial(5, 2, 2, 1, 1) → 83160 return prod(map(comb, accumulate(counts), counts))

def powerset(iterable): "Subsequences of the iterable from shortest to longest." # 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 sum_of_squares(iterable): "Add up the squares of the input values." # sum_of_squares([10, 20, 30]) → 1400 return sumprod(*tee(iterable))

==== Matrix operations ====

def reshape(matrix, columns): "Reshape a 2-D matrix to have a given number of columns." # reshape([(0, 1), (2, 3), (4, 5)], 3) → (0, 1, 2) (3, 4, 5) return batched(chain.from_iterable(matrix), columns, strict=True)

def transpose(matrix): "Swap the rows and columns of a 2-D matrix." # transpose([(1, 2, 3), (11, 22, 33)]) → (1, 11) (2, 22) (3, 33) return zip(*matrix, 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)

==== Polynomial arithmetic ====

def convolve(signal, kernel): """Discrete linear convolution of two iterables. Equivalent to polynomial multiplication.

   Convolutions are mathematically commutative; however, the inputs are
   evaluated differently.  The signal is consumed lazily and can be
   infinite. The kernel is fully consumed before the calculations begin.

   Article:  https://betterexplained.com/articles/intuitive-convolution/
   Video:    https://www.youtube.com/watch?v=KuXjwB4LzSA
   """
   # convolve([1, -1, -20], [1, -3]) → 1 -4 -17 60
   # convolve(data, [0.25, 0.25, 0.25, 0.25]) → Moving average (blur)
   # convolve(data, [1/2, 0, -1/2]) → 1st derivative estimate
   # convolve(data, [1, -2, 1]) → 2nd derivative estimate
   kernel = tuple(kernel)[::-1]
   n = len(kernel)
   padded_signal = chain(repeat(0, n-1), signal, repeat(0, n-1))
   windowed_signal = sliding_window(padded_signal, n)
   return map(sumprod, repeat(kernel), windowed_signal)

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]
   factors = zip(repeat(1), map(neg, roots))
   return list(reduce(convolve, factors, [1]))

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 = 5
   # polynomial_eval([1, -4, -17, 60], x=5) → 0
   n = len(coefficients)
   if not n:
       return type(x)(0)
   powers = map(pow, repeat(x), reversed(range(n)))
   return sumprod(coefficients, powers)

def polynomial_derivative(coefficients): """Compute the first derivative of a polynomial.

      f(x)  =  x³ -4x² -17x + 60
      f'(x) = 3x² -8x  -17
   """
   # polynomial_derivative([1, -4, -17, 60]) → [3, -8, -17]
   n = len(coefficients)
   powers = reversed(range(1, n))
   return list(map(mul, coefficients, powers))

==== Number theory ====

def sieve(n): "Primes less than n." # sieve(30) → 2 3 5 7 11 13 17 19 23 29 if n > 2: yield 2 data = bytearray((0, 1)) * (n // 2) for p in iter_index(data, 1, start=3, stop=isqrt(n) + 1): data[pp : n : p+p] = bytes(len(range(pp, n, p+p))) yield from iter_index(data, 1, start=3)

def factor(n): "Prime factors of n." # factor(99) → 3 3 11 # factor(1_000_000_000_000_007) → 47 59 360620266859 # factor(1_000_000_000_000_403) → 1000000000000403 for prime in sieve(isqrt(n) + 1): while not n % prime: yield prime n //= prime if n == 1: return if n > 1: yield n

def is_prime(n): "Return True if n is prime." # is_prime(1_000_000_000_000_403) → True return n > 1 and next(factor(n)) == n

def totient(n): "Count of natural numbers up to n that are coprime to n." # https://mathworld.wolfram.com/TotientFunction.html # totient(12) → 4 because len([1, 5, 7, 11]) == 4 for prime in set(factor(n)): n -= n // prime return n

.. doctest:: :hide:

These examples no longer appear in the docs but are guaranteed
to keep working.

>>> amounts = [120.15, 764.05, 823.14]
>>> for checknum, amount in zip(count(1200), amounts):
...     print('Check %d is for $%.2f' % (checknum, amount))
...
Check 1200 is for $120.15
Check 1201 is for $764.05
Check 1202 is for $823.14

>>> import operator
>>> for cube in map(operator.pow, range(1,4), repeat(3)):
...    print(cube)
...
1
8
27

>>> reportlines = ['EuroPython', 'Roster', '', 'alex', '', 'laura', '', 'martin', '', 'walter', '', 'samuele']
>>> for name in islice(reportlines, 3, None, 2):
...    print(name.title())
...
Alex
Laura
Martin
Walter
Samuele

>>> from operator import itemgetter
>>> d = dict(a=1, b=2, c=1, d=2, e=1, f=2, g=3)
>>> di = sorted(sorted(d.items()), key=itemgetter(1))
>>> for k, g in groupby(di, itemgetter(1)):
...     print(k, list(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(list(map(operator.itemgetter(1), g)))
...
[1]
[4, 5, 6]
[10]
[15, 16, 17, 18]
[22]
[25, 26, 27, 28]

Now, we test all of the itertool recipes

>>> take(10, count())
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
>>> # Verify that the input is consumed lazily
>>> it = iter('abcdef')
>>> take(3, it)
['a', 'b', 'c']
>>> list(it)
['d', 'e', 'f']


>>> list(prepend(1, [2, 3, 4]))
[1, 2, 3, 4]


>>> list(enumerate('abc'))
[(0, 'a'), (1, 'b'), (2, 'c')]


>>> list(running_mean([8.5, 9.5, 7.5, 6.5]))
[8.5, 9.0, 8.5, 8.0]


>>> for _ in loops(5):
...     print('hi')
...
hi
hi
hi
hi
hi


>>> list(tail(3, 'ABCDEFG'))
['E', 'F', 'G']
>>> # Verify the input is consumed greedily
>>> input_iterator = iter('ABCDEFG')
>>> output_iterator = tail(3, input_iterator)
>>> list(input_iterator)
[]


>>> it = iter(range(10))
>>> consume(it, 3)
>>> # Verify the input is consumed lazily
>>> next(it)
3
>>> # Verify the input is consumed completely
>>> consume(it)
>>> next(it, 'Done')
'Done'


>>> nth('abcde', 3)
'd'
>>> nth('abcde', 9) is None
True
>>> # Verify that the input is consumed lazily
>>> it = iter('abcde')
>>> nth(it, 2)
'c'
>>> list(it)
['d', 'e']


>>> [all_equal(s) for s in ('', 'A', 'AAAA', 'AAAB', 'AAABA')]
[True, True, True, False, False]
>>> [all_equal(s, key=str.casefold) for s in ('', 'A', 'AaAa', 'AAAB', 'AAABA')]
[True, True, True, False, False]
>>> # Verify that the input is consumed lazily and that only
>>> # one element of a second equivalence class is used to disprove
>>> # the assertion that all elements are equal.
>>> it = iter('aaabbbccc')
>>> all_equal(it)
False
>>> ''.join(it)
'bbccc'


>>> quantify(range(99), lambda x: x%2==0)
50
>>> quantify([True, False, False, True, True])
3
>>> quantify(range(12), predicate=lambda x: x%2==1)
6


>>> a = [[1, 2, 3], [4, 5, 6]]
>>> list(flatten(a))
[1, 2, 3, 4, 5, 6]


>>> list(ncycles('abc', 3))
['a', 'b', 'c', 'a', 'b', 'c', 'a', 'b', 'c']
>>> # Verify greedy consumption of input iterator
>>> input_iterator = iter('abc')
>>> output_iterator = ncycles(input_iterator, 3)
>>> list(input_iterator)
[]


>>> sum_of_squares([10, 20, 30])
1400


>>> list(reshape([(0, 1), (2, 3), (4, 5)], 3))
[(0, 1, 2), (3, 4, 5)]
>>> M = [(0, 1, 2, 3), (4, 5, 6, 7), (8, 9, 10, 11)]
>>> list(reshape(M, 1))
[(0,), (1,), (2,), (3,), (4,), (5,), (6,), (7,), (8,), (9,), (10,), (11,)]
>>> list(reshape(M, 2))
[(0, 1), (2, 3), (4, 5), (6, 7), (8, 9), (10, 11)]
>>> list(reshape(M, 3))
[(0, 1, 2), (3, 4, 5), (6, 7, 8), (9, 10, 11)]
>>> list(reshape(M, 4))
[(0, 1, 2, 3), (4, 5, 6, 7), (8, 9, 10, 11)]
>>> list(reshape(M, 5))
Traceback (most recent call last):
...
ValueError: batched(): incomplete batch
>>> list(reshape(M, 6))
[(0, 1, 2, 3, 4, 5), (6, 7, 8, 9, 10, 11)]
>>> list(reshape(M, 12))
[(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11)]


>>> list(transpose([(1, 2, 3), (11, 22, 33)]))
[(1, 11), (2, 22), (3, 33)]
>>> # Verify that the inputs are consumed lazily
>>> input1 = iter([1, 2, 3])
>>> input2 = iter([11, 22, 33])
>>> output_iterator = transpose([input1, input2])
>>> next(output_iterator)
(1, 11)
>>> list(zip(input1, input2))
[(2, 22), (3, 33)]


>>> list(matmul([(7, 5), (3, 5)], [[2, 5], [7, 9]]))
[(49, 80), (41, 60)]
>>> list(matmul([[2, 5], [7, 9], [3, 4]], [[7, 11, 5, 4, 9], [3, 5, 2, 6, 3]]))
[(29, 47, 20, 38, 33), (76, 122, 53, 82, 90), (33, 53, 23, 36, 39)]


>>> list(convolve([1, -1, -20], [1, -3])) == [1, -4, -17, 60]
True
>>> data = [20, 40, 24, 32, 20, 28, 16]
>>> list(convolve(data, [0.25, 0.25, 0.25, 0.25]))
[5.0, 15.0, 21.0, 29.0, 29.0, 26.0, 24.0, 16.0, 11.0, 4.0]
>>> list(convolve(data, [1, -1]))
[20, 20, -16, 8, -12, 8, -12, -16]
>>> list(convolve(data, [1, -2, 1]))
[20, 0, -36, 24, -20, 20, -20, -4, 16]
>>> # Verify signal is consumed lazily and the kernel greedily
>>> signal_iterator = iter([10, 20, 30, 40, 50])
>>> kernel_iterator = iter([1, 2, 3])
>>> output_iterator = convolve(signal_iterator, kernel_iterator)
>>> list(kernel_iterator)
[]
>>> next(output_iterator)
10
>>> next(output_iterator)
40
>>> list(signal_iterator)
[30, 40, 50]


>>> from fractions import Fraction
>>> from decimal import Decimal
>>> polynomial_eval([1, -4, -17, 60], x=5)
0
>>> x = 5; x**3 - 4*x**2 -17*x + 60
0
>>> polynomial_eval([1, -4, -17, 60], x=2.5)
8.125
>>> x = 2.5; x**3 - 4*x**2 -17*x + 60
8.125
>>> polynomial_eval([1, -4, -17, 60], x=Fraction(2, 3))
Fraction(1274, 27)
>>> x = Fraction(2, 3); x**3 - 4*x**2 -17*x + 60
Fraction(1274, 27)
>>> polynomial_eval([1, -4, -17, 60], x=Decimal('1.75'))
Decimal('23.359375')
>>> x = Decimal('1.75'); x**3 - 4*x**2 -17*x + 60
Decimal('23.359375')
>>> polynomial_eval([], 2)
0
>>> polynomial_eval([], 2.5)
0.0
>>> polynomial_eval([], Fraction(2, 3))
Fraction(0, 1)
>>> polynomial_eval([], Decimal('1.75'))
Decimal('0')
>>> polynomial_eval([11], 7) == 11
True
>>> polynomial_eval([11, 2], 7) == 11 * 7 + 2
True


>>> polynomial_from_roots([5, -4, 3])
[1, -4, -17, 60]
>>> factored = lambda x: (x - 5) * (x + 4) * (x - 3)
>>> expanded = lambda x: x**3 -4*x**2 -17*x + 60
>>> all(factored(x) == expanded(x) for x in range(-10, 11))
True


>>> polynomial_derivative([1, -4, -17, 60])
[3, -8, -17]


>>> list(iter_index('AABCADEAF', 'A'))
[0, 1, 4, 7]
>>> list(iter_index('AABCADEAF', 'B'))
[2]
>>> list(iter_index('AABCADEAF', 'X'))
[]
>>> list(iter_index('', 'X'))
[]
>>> list(iter_index('AABCADEAF', 'A', 1))
[1, 4, 7]
>>> list(iter_index(iter('AABCADEAF'), 'A', 1))
[1, 4, 7]
>>> list(iter_index('AABCADEAF', 'A', 2))
[4, 7]
>>> list(iter_index(iter('AABCADEAF'), 'A', 2))
[4, 7]
>>> list(iter_index('AABCADEAF', 'A', 10))
[]
>>> list(iter_index(iter('AABCADEAF'), 'A', 10))
[]
>>> list(iter_index('AABCADEAF', 'A', 1, 7))
[1, 4]
>>> list(iter_index(iter('AABCADEAF'), 'A', 1, 7))
[1, 4]
>>> # Verify that ValueErrors not swallowed (gh-107208)
>>> def assert_no_value(iterable, forbidden_value):
...     for item in iterable:
...         if item == forbidden_value:
...             raise ValueError
...         yield item
...
>>> list(iter_index(assert_no_value('AABCADEAF', 'B'), 'A'))
Traceback (most recent call last):
...
ValueError
>>> # Verify that both paths can find identical NaN values
>>> x = float('NaN')
>>> y = float('NaN')
>>> list(iter_index([0, x, x, y, 0], x))
[1, 2]
>>> list(iter_index(iter([0, x, x, y, 0]), x))
[1, 2]
>>> # Test list input. Lists do not support None for the stop argument
>>> list(iter_index(list('AABCADEAF'), 'A'))
[0, 1, 4, 7]
>>> # Verify that input is consumed lazily
>>> input_iterator = iter('AABCADEAF')
>>> output_iterator = iter_index(input_iterator, 'A')
>>> next(output_iterator)
0
>>> next(output_iterator)
1
>>> next(output_iterator)
4
>>> ''.join(input_iterator)
'DEAF'


>>> # Verify that the target value can be a sequence.
>>> seq = [[10, 20], [30, 40], 30, 40, [30, 40], 50]
>>> target = [30, 40]
>>> list(iter_index(seq, target))
[1, 4]


>>> # Verify faithfulness to type specific index() method behaviors.
>>> # For example, bytes and str perform continuous-subsequence searches
>>> # that do not match the general behavior specified
>>> # in collections.abc.Sequence.index().
>>> seq = 'abracadabra'
>>> target = 'ab'
>>> list(iter_index(seq, target))
[0, 7]


>>> list(sieve(30))
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
>>> small_primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
>>> all(list(sieve(n)) == [p for p in small_primes if p < n] for n in range(101))
True
>>> len(list(sieve(100)))
25
>>> len(list(sieve(1_000)))
168
>>> len(list(sieve(10_000)))
1229
>>> len(list(sieve(100_000)))
9592
>>> len(list(sieve(1_000_000)))
78498
>>> carmichael = {561, 1105, 1729, 2465, 2821, 6601, 8911}  # https://oeis.org/A002997
>>> set(sieve(10_000)).isdisjoint(carmichael)
True


>>> small_primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
>>> list(filter(is_prime, range(-100, 100))) == small_primes
True
>>> carmichael = {561, 1105, 1729, 2465, 2821, 6601, 8911}  # https://oeis.org/A002997
>>> any(map(is_prime, carmichael))
False
>>> # https://www.wolframalpha.com/input?i=is+128884753939+prime
>>> is_prime(128_884_753_939)           # large prime
True
>>> is_prime(999953 * 999983)           # large semiprime
False
>>> is_prime(1_000_000_000_000_007)     # factor() example
False
>>> is_prime(1_000_000_000_000_403)     # factor() example
True


>>> list(factor(99))                    # Code example 1
[3, 3, 11]
>>> list(factor(1_000_000_000_000_007)) # Code example 2
[47, 59, 360620266859]
>>> list(factor(1_000_000_000_000_403)) # Code example 3
[1000000000000403]
>>> list(factor(0))
[]
>>> list(factor(1))
[]
>>> list(factor(2))
[2]
>>> list(factor(3))
[3]
>>> list(factor(4))
[2, 2]
>>> list(factor(5))
[5]
>>> list(factor(6))
[2, 3]
>>> list(factor(7))
[7]
>>> list(factor(8))
[2, 2, 2]
>>> list(factor(9))
[3, 3]
>>> list(factor(10))
[2, 5]
>>> list(factor(128_884_753_939))       # large prime
[128884753939]
>>> list(factor(999953 * 999983))       # large semiprime
[999953, 999983]
>>> list(factor(6 ** 20)) == [2] * 20 + [3] * 20   # large power
True
>>> list(factor(909_909_090_909))       # large multiterm composite
[3, 3, 7, 13, 13, 751, 113797]
>>> math.prod([3, 3, 7, 13, 13, 751, 113797])
909909090909
>>> all(math.prod(factor(n)) == n for n in range(1, 2_000))
True
>>> all(set(factor(n)) <= set(sieve(n+1)) for n in range(2_000))
True
>>> all(list(factor(n)) == sorted(factor(n)) for n in range(2_000))
True


>>> totient(0)  # https://www.wolframalpha.com/input?i=totient+0
0
>>> first_totients = [1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6,
... 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30, 16, 20, 16, 24, 12, 36, 18,
... 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 42, 20, 32, 24, 52, 18, 40, 24, 36,
... 28, 58, 16, 60, 30, 36, 32, 48, 20, 66, 32, 44]  # https://oeis.org/A000010
...
>>> list(map(totient, range(1, 70))) == first_totients
True
>>> reference_totient = lambda n: sum(math.gcd(t, n) == 1 for t in range(1, n+1))
>>> all(totient(n) == reference_totient(n) for n in range(1000))
True
>>> totient(128_884_753_939) == 128_884_753_938  # large prime
True
>>> totient(999953 * 999983) == 999952 * 999982  # large semiprime
True
>>> totient(6 ** 20) == 1 * 2**19 * 2 * 3**19    # repeated primes
True


>>> list(flatten([('a', 'b'), (), ('c', 'd', 'e'), ('f',), ('g', 'h', 'i')]))
['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i']


>>> list(repeatfunc(pow, 5, 2, 3))
[8, 8, 8, 8, 8]
>>> take(5, map(int, repeatfunc(random.random)))
[0, 0, 0, 0, 0]
>>> random.seed(85753098575309)
>>> list(repeatfunc(random.random, 3))
[0.16370491282496968, 0.45889608687313455, 0.3747076837820118]
>>> list(repeatfunc(chr, 3, 65))
['A', 'A', 'A']
>>> list(repeatfunc(pow, 3, 2, 5))
[32, 32, 32]


>>> list(grouper('abcdefg', 3, fillvalue='x'))
[('a', 'b', 'c'), ('d', 'e', 'f'), ('g', 'x', 'x')]


>>> it = grouper('abcdefg', 3, incomplete='strict')
>>> next(it)
('a', 'b', 'c')
>>> next(it)
('d', 'e', 'f')
>>> next(it)
Traceback (most recent call last):
  ...
ValueError: zip() argument 2 is shorter than argument 1

>>> list(grouper('abcdefg', n=3, incomplete='ignore'))
[('a', 'b', 'c'), ('d', 'e', 'f')]


>>> list(sliding_window('ABCDEFG', 1))
[('A',), ('B',), ('C',), ('D',), ('E',), ('F',), ('G',)]
>>> list(sliding_window('ABCDEFG', 2))
[('A', 'B'), ('B', 'C'), ('C', 'D'), ('D', 'E'), ('E', 'F'), ('F', 'G')]
>>> list(sliding_window('ABCDEFG', 3))
[('A', 'B', 'C'), ('B', 'C', 'D'), ('C', 'D', 'E'), ('D', 'E', 'F'), ('E', 'F', 'G')]
>>> list(sliding_window('ABCDEFG', 4))
[('A', 'B', 'C', 'D'), ('B', 'C', 'D', 'E'), ('C', 'D', 'E', 'F'), ('D', 'E', 'F', 'G')]
>>> list(sliding_window('ABCDEFG', 5))
[('A', 'B', 'C', 'D', 'E'), ('B', 'C', 'D', 'E', 'F'), ('C', 'D', 'E', 'F', 'G')]
>>> list(sliding_window('ABCDEFG', 6))
[('A', 'B', 'C', 'D', 'E', 'F'), ('B', 'C', 'D', 'E', 'F', 'G')]
>>> list(sliding_window('ABCDEFG', 7))
[('A', 'B', 'C', 'D', 'E', 'F', 'G')]
>>> list(sliding_window('ABCDEFG', 8))
[]
>>> try:
...     list(sliding_window('ABCDEFG', -1))
... except ValueError:
...     'zero or negative n not supported'
...
'zero or negative n not supported'
>>> try:
...     list(sliding_window('ABCDEFG', 0))
... except ValueError:
...     'zero or negative n not supported'
...
'zero or negative n not supported'


>>> list(roundrobin('abc', 'd', 'ef'))
['a', 'd', 'e', 'b', 'f', 'c']
>>> ranges = [range(5, 1000), range(4, 3000), range(0), range(3, 2000), range(2, 5000), range(1, 3500)]
>>> collections.Counter(roundrobin(*ranges)) == collections.Counter(chain(*ranges))
True
>>> # Verify that the inputs are consumed lazily
>>> input_iterators = list(map(iter, ['abcd', 'ef', '', 'ghijk', 'l', 'mnopqr']))
>>> output_iterator = roundrobin(*input_iterators)
>>> ''.join(islice(output_iterator, 10))
'aeglmbfhnc'
>>> ''.join(chain(*input_iterators))
'dijkopqr'


>>> list(subslices('ABCD'))
['A', 'AB', 'ABC', 'ABCD', 'B', 'BC', 'BCD', 'C', 'CD', 'D']


>>> ' '.join(map(''.join, derangements('ABCD')))
'BADC BCDA BDAC CADB CDAB CDBA DABC DCAB DCBA'
>>> ' '.join(map(''.join, derangements('ABCD', 3)))
'BAD BCA BCD BDA CAB CAD CDA CDB DAB DCA DCB'
>>> ' '.join(map(''.join, derangements('ABCD', 2)))
'BA BC BD CA CD DA DC'
>>> ' '.join(map(''.join, derangements('ABCD', 1)))
'B C D'
>>> ' '.join(map(''.join, derangements('ABCD', 0)))
''
>>> # Compare number of derangements to https://oeis.org/A000166
>>> [len(list(derangements(range(n)))) for n in range(10)]
[1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496]
>>> # Verify that identical objects are treated as unique by position
>>> identical = 'X'
>>> distinct = 'x'
>>> seq1 = ('A', identical, 'B', identical)
>>> result1 = ' '.join(map(''.join, derangements(seq1)))
>>> result1
'XAXB XBXA XXAB BAXX BXAX BXXA XAXB XBAX XBXA'
>>> seq2 = ('A', identical, 'B', distinct)
>>> result2 = ' '.join(map(''.join, derangements(seq2)))
>>> result2
'XAxB XBxA XxAB BAxX BxAX BxXA xAXB xBAX xBXA'
>>> result1 == result2
False
>>> result1.casefold() == result2.casefold()
True


>>> list(powerset([1,2,3]))
[(), (1,), (2,), (3,), (1, 2), (1, 3), (2, 3), (1, 2, 3)]
>>> all(len(list(powerset(range(n)))) == 2**n for n in range(18))
True
>>> list(powerset('abcde')) == sorted(sorted(set(powerset('abcde'))), key=len)
True


>>> list(unique_everseen('AAAABBBCCDAABBB'))
['A', 'B', 'C', 'D']
>>> list(unique_everseen('ABBCcAD', str.casefold))
['A', 'B', 'C', 'D']
>>> list(unique_everseen('ABBcCAD', str.casefold))
['A', 'B', 'c', 'D']
>>> # Verify that the input is consumed lazily
>>> input_iterator = iter('AAAABBBCCDAABBB')
>>> output_iterator = unique_everseen(input_iterator)
>>> next(output_iterator)
'A'
>>> ''.join(input_iterator)
'AAABBBCCDAABBB'


>>> list(unique_justseen('AAAABBBCCDAABBB'))
['A', 'B', 'C', 'D', 'A', 'B']
>>> list(unique_justseen('ABBCcAD', str.casefold))
['A', 'B', 'C', 'A', 'D']
>>> list(unique_justseen('ABBcCAD', str.casefold))
['A', 'B', 'c', 'A', 'D']
>>> # Verify that the input is consumed lazily
>>> input_iterator = iter('AAAABBBCCDAABBB')
>>> output_iterator = unique_justseen(input_iterator)
>>> next(output_iterator)
'A'
>>> ''.join(input_iterator)
'AAABBBCCDAABBB'


>>> list(unique([[1, 2], [3, 4], [1, 2]]))
[[1, 2], [3, 4]]
>>> list(unique('ABBcCAD', str.casefold))
['A', 'B', 'c', 'D']
>>> list(unique('ABBcCAD', str.casefold, reverse=True))
['D', 'c', 'B', 'A']


>>> d = dict(a=1, b=2, c=3)
>>> it = iter_except(d.popitem, KeyError)
>>> d['d'] = 4
>>> next(it)
('d', 4)
>>> next(it)
('c', 3)
>>> next(it)
('b', 2)
>>> d['e'] = 5
>>> next(it)
('e', 5)
>>> next(it)
('a', 1)
>>> next(it, 'empty')
'empty'


>>> first_true('ABC0DEF1', '9', str.isdigit)
'0'
>>> # Verify that inputs are consumed lazily
>>> it = iter('ABC0DEF1')
>>> first_true(it, predicate=str.isdigit)
'0'
>>> ''.join(it)
'DEF1'

>>> multinomial(5, 2, 2, 1, 1)
83160
>>> word = 'coffee'
>>> multinomial(*Counter(word).values()) == len(set(permutations(word)))
True

.. testcode:: :hide:

# Old recipes and their tests which are guaranteed to continue to work.

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

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

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

def pad_none(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 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 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)

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 true iterator must be fully consumed
       before the remainder 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

   return true_iterator(), chain(transition, it)

def partition(predicate, iterable):
    """Partition entries into false entries and true entries.

    If *predicate* is slow, consider wrapping it with functools.lru_cache().
    """
    # partition(is_odd, range(10)) → 0 2 4 6 8   and  1 3 5 7 9
    t1, t2 = tee(iterable)
    return filterfalse(predicate, t1), filter(predicate, t2)

.. doctest:: :hide:

>>> list(islice(tabulate(lambda x: 2*x), 4))
[0, 2, 4, 6]


>>> dotproduct([1,2,3], [4,5,6])
32


>>> old_sumprod_recipe([1,2,3], [4,5,6])
32


>>> list(islice(pad_none('abc'), 0, 6))
['a', 'b', 'c', None, None, None]


>>> list(triplewise('ABCDEFG'))
[('A', 'B', 'C'), ('B', 'C', 'D'), ('C', 'D', 'E'), ('D', 'E', 'F'), ('E', 'F', 'G')]


>>> population = 'ABCDEFGH'
>>> for r in range(len(population) + 1):
...     seq = list(combinations(population, r))
...     for i in range(len(seq)):
...         assert nth_combination(population, r, i) == seq[i]
...     for i in range(-len(seq), 0):
...         assert nth_combination(population, r, i) == seq[i]
...
>>> iterable = 'abcde'
>>> r = 3
>>> combos = list(combinations(iterable, r))
>>> all(nth_combination(iterable, r, i) == comb for i, comb in enumerate(combos))
True


>>> it = iter('ABCdEfGhI')
>>> all_upper, remainder = before_and_after(str.isupper, it)
>>> ''.join(all_upper)
'ABC'
>>> ''.join(remainder)
'dEfGhI'


>>> def is_odd(x):
...     return x % 2 == 1
...
>>> evens, odds = partition(is_odd, range(10))
>>> list(evens)
[0, 2, 4, 6, 8]
>>> list(odds)
[1, 3, 5, 7, 9]
>>> # Verify that the input is consumed lazily
>>> input_iterator = iter(range(10))
>>> evens, odds = partition(is_odd, input_iterator)
>>> next(odds)
1
>>> next(odds)
3
>>> next(evens)
0
>>> list(input_iterator)
[4, 5, 6, 7, 8, 9]

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

:mod:!itertools --- Functions creating iterators for efficient looping

Itertool Functions

Itertools Recipes

==== Basic one liners ====

==== Data pipelines ====

==== Mathematical operations ====

==== Matrix operations ====

==== Polynomial arithmetic ====

==== Number theory ====

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