itertools — Functions creating iterators for efficient looping


Esse módulo implementa diversos blocos de instruções com iteradores, inspirados por construções de APL, Haskell, e SML. Cada uma foi adequadamente reformulada para Python.

Esse módulo padroniza um conjunto central de ferramentas rápidas e de uso eficiente da memória, que podem ser utilizadas sozinhas ou combinadas. Juntas, eles formam uma “álgebra de iteradores” tornando possível construir ferramentas sucintas e eficientes em Python puro.

Por exemplo, SML fornece uma ferramenta para tabulação: tabulate(f) que produz uma sequência f(0), f(1), .... O mesmo efeito pode ser obtido em Python combinando map() e count() para formar 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))).

Iteradores infinitos:

Iterador

Argumentos

Resultado

Exemplo

count()

[start[, step]]

start, start+step, start+2*step, …

count(10) 10 11 12 13 14 ...

cycle()

p

p0, p1, … ultimo elemento de p, p0, p1, …

cycle('ABCD') A B C D A B C D ...

repeat()

elem [,n]

elem, elem, elem, … repete infinitamente ou até n vezes

repeat(10, 3) 10 10 10

Iteradores terminando na sequencia de entrada mais curta:

Iterador

Argumentos

Resultado

Exemplo

accumulate()

p [,func]

p0, p0+p1, p0+p1+p2, …

accumulate([1,2,3,4,5]) 1 3 6 10 15

batched()

p, n

(p0, p1, …, p_n-1), …

batched('ABCDEFG', n=3) ABC DEF G

chain()

p, q, …

p0, p1, … último elemento de p, q0, q1, …

chain('ABC', 'DEF') A B C D E F

chain.from_iterable()

iterable

p0, p1, … último elemento de p, 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()

predicate, seq

seq[n], seq[n+1], starting when predicate fails

dropwhile(lambda x: x<5, [1,4,6,4,1]) 6 4 1

filterfalse()

predicate, seq

elements of seq where predicate(elem) fails

filterfalse(lambda x: x%2, range(10)) 0 2 4 6 8

groupby()

iterable[, key]

sub-iteradores agrupados pelo valor de key(v)

islice()

seq, [start,] stop [, step]

elementos de 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()

predicate, seq

seq[0], seq[1], until predicate fails

takewhile(lambda x: x<5, [1,4,6,4,1]) 1 4

tee()

it, n

n iteradores it independentes

zip_longest()

p, q, …

(p[0], q[0]), (p[1], q[1]), …

zip_longest('ABCD', 'xy', fillvalue='-') Ax By C- D-

Iteradores combinatórios:

Iterador

Argumentos

Resultado

product()

p, q, … [repeat=1]

produto cartesiano, equivalente a laços for aninhados

permutations()

p[, r]

tuplas de tamanho r, com todas ordenações possíveis, sem elementos repetidos

combinations()

p, r

tuplas de tamanho r, ordenadas, sem elementos repetidos

combinations_with_replacement()

p, r

tuplas de tamanho r, ordenadas, com elementos repetidos

Exemplos

Resultado

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

Todas as funções a seguir constroem e retorna iteradores. Algumas fornecem fluxos de tamanhos infinitos, assim elas devem ser acessados somente por funções ou laços que interrompem o fluxo.

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

Constrói um iterador que devolve somas acumuladas, ou resultados acumulados de outras funções (especificada pelo argumento opcional func).

Se func é fornecido, deve ser uma função de dois argumentos. Elementos da entrada iterable pode ser qualquer tipo que pode ser aceito como argumento para func. (Por exemplo, com o operador padrão de adição, esses elementos pode ser qualquer tipo que possa ser somado, incluindo Decimal ou Fraction.)

Usualmente, o número de elementos da saída coincide com o número de elementos do iterável da entrada. Contudo, se o argumento nomeado initial é fornecido, a acumulação começa com este valor inicial, assim a saída terá um elemento a mais que o iterável da entrada.

Aproximadamente equivalente a:

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
    iterator = iter(iterable)
    total = initial
    if initial is None:
        try:
            total = next(iterator)
        except StopIteration:
            return
    yield total
    for element in iterator:
        total = func(total, element)
        yield total

The func argument 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 10 annual payments of 90
>>> account_update = lambda bal, pmt: round(bal * 1.05) + pmt
>>> list(accumulate(repeat(-90, 10), account_update, initial=1_000))
[1000, 960, 918, 874, 828, 779, 728, 674, 618, 559, 497]

Veja functools.reduce() para uma função similar que devolve apenas o valor acumulado final.

Adicionado na versão 3.2.

Alterado na versão 3.3: Adicionado o parâmetro opcional func.

Alterado na versão 3.8: Adicionado o parâmetro opcional initial.

itertools.batched(iterable, n)

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

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

Aproximadamente equivalente a:

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

Adicionado na versão 3.12.

itertools.chain(*iterables)

Cria um iterador que devolve elementos do primeiro iterável até o esgotamento, então continua com o próximo iterável, até que todos os iteráveis sejam esgotados. Usando para tratar sequências consecutivas como uma única sequencia. aproximadamente equivalente a:

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

Construtor alternativo para chain(). Obtém entradas encadeadas a partir de um único argumento iterável que avaliado preguiçosamente. Aproximadamente equivalente a:

def from_iterable(iterables):
    # chain.from_iterable(['ABC', 'DEF']) → A B C D E F
    for iterable in iterables:
        yield from iterable
itertools.combinations(iterable, r)

Devolve subsequências de elementos com comprimento r a partir da entrada iterável

The combination 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 each combination.

Aproximadamente equivalente a:

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)

O código para combinations() também pode ser expresso como uma subsequência de permutations() depois de filtradas as entradas onde os elementos não estão ordenandos (de acordo com a sua posição na entrada):

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)

O número de itens devolvidos é n! / r! / (n-r)! quando 0 <= r <= n ou zero quando r > n.

itertools.combinations_with_replacement(iterable, r)

Devolve subsequências de comprimento r de elementos do iterável de entrada permitindo que elementos individuais sejam repetidos mais de uma vez.

The combination 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, the generated combinations will also be unique.

Aproximadamente equivalente a:

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)

O código para combinations_with_replacement() também pode ser expresso como uma subsequência de product() depois de filtradas as entradas onde os elementos não estão ordenados (de acordo com a sua posição na entrada):

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)

O número de itens devolvidos é (n+r-1)! / r! / (n-1)! quando n > 0.

Adicionado na versão 3.1.

itertools.compress(data, selectors)

Make an iterator that filters elements from data returning only those that have a 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)

Adicionado na versão 3.1.

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

Crie um iterador que devolve valores igualmente espaçados começando pelo número start. Frequentemente usado com um argumento da função map() para gerar pontos de dados consecutivos. Também usado com zip() para adicionar números sequenciais. Aproximadamente equivalente a:

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

Quando é feita uma contagem usando números de ponto flutuante, é possível ter melhor precisão substituindo código multiplicativo como (start + step * i for i in count()).

Alterado na versão 3.1: Adicionou argumento step e permitiu argumentos não-inteiros.

itertools.cycle(iterable)

Crie um iterador que devolve elementos do iterável assim como salva uma cópia de cada um. Quando o iterável é esgotado, devolve elementos da cópia salva. Repete indefinidamente. Aproximadamente equivalente a:

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, [1,4,6,3,8]) → 6 3 8
    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<5, [1,4,6,3,8]) → 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() é aproximadamente equivalente a:

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
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).

Aproximadamente equivalente a:

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
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.

Aproximadamente equivalente a:

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

Adicionado na versão 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.

Aproximadamente equivalente a:

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)

O número de itens retornado é n! / (n-r)! quando 0 <= r <= n ou zero quando r > n.

itertools.product(*iterables, repeat=1)

Produto cartesiano de iteráveis de entrada

Aproximadamente equivalente a laços for aninhados em uma expressão geradora. Por exemplo, product(A, B) devolve o mesmo que ((x,y) for x in A for y in B).

Os laços aninhados circulam como um hodômetro com o elemento mais à direita avançando a cada iteração. Este padrão cria uma ordenação lexicográfica de maneira que se os iteráveis de entrada estiverem ordenados, as tuplas produzidas são emitidas de maneira ordenada.

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
    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 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.

Aproximadamente equivalente a:

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

Um uso comum de repeat é para fornecer um fluxo de valores constantes para map ou 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, [1,4,6,3,8]) → 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-iterools before_and_after() instead.

itertools.tee(iterable, n=2)

Return n independent iterators from a single iterable.

Aproximadamente equivalente a:

def tee(iterable, n=2):
    iterator = iter(iterable)
    shared_link = [None, None]
    return tuple(_tee(iterator, shared_link) for _ in range(n))

def _tee(iterator, link):
    try:
        while True:
            if link[1] is None:
                link[0] = next(iterator)
                link[1] = [None, None]
            value, link = link
            yield value
    except StopIteration:
        return

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 simultaneously using 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(*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)

Se um dos iteráveis é potencialmente infinito, então a função zip_longest() deve ser embrulhada por algo que limite o número de chamadas (por exemplo islice() ou takewhile()). Se não especificado, fillvalue tem o valor padrão None.

Receitas com itertools

Esta seção mostra receitas para criação de um ferramental ampliado usando as ferramentas existentes de itertools como elementos construtivos.

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 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 sliding_window(), iter_index(), 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 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. High speed is retained by preferring “vectorized” building blocks over the use of for-loops and generators which incur interpreter overhead.

import collections
import contextlib
import functools
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 repeatfunc(func, times=None, *args):
    "Repeat calls to func with specified arguments."
    if times is None:
        return starmap(func, repeat(args))
    return starmap(func, 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 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:
        collections.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

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.casefold) → A B c A D
    if key is None:
        return map(operator.itemgetter(0), groupby(iterable))
    return map(next, map(operator.itemgetter(1), groupby(iterable, key)))

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.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 sliding_window(iterable, n):
    "Collect data into overlapping fixed-length chunks or blocks."
    # sliding_window('ABCDEFG', 4) → ABCD BCDE CDEF DEFG
    iterator = iter(iterable)
    window = collections.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 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)

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 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 contextlib.suppress(ValueError):
            while True:
                yield (i := seq_index(value, i, stop))
                i += 1

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

The following recipes have a more mathematical flavor:

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 sum_of_squares(iterable):
    "Add up the squares of the input values."
    # sum_of_squares([10, 20, 30]) → 1400
    return math.sumprod(*tee(iterable))

def reshape(matrix, cols):
    "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), cols)

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(math.sumprod, product(m1, transpose(m2))), n)

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(math.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(operator.neg, roots))
    return list(functools.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 math.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(operator.mul, coefficients, powers))

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=math.isqrt(n) + 1):
        data[p*p : n : p+p] = bytes(len(range(p*p, 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(math.isqrt(n) + 1):
        while not n % prime:
            yield prime
            n //= prime
            if n == 1:
                return
    if n > 1:
        yield 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