itertools — Functions creating iterators for efficient looping

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

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

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

These tools and their built-in counterparts also work well with the high-speed functions in the operator module. For example, the multiplication operator can be mapped across two vectors to form an efficient dot-product: sum(starmap(operator.mul, zip(vec1, vec2, strict=True))).

Infinite iterators:

Iterator	Arguments	Results	Example
count()	start, [step]	start, start+step, start+2*step, …	count(10) --> 10 11 12 13 14 ...
cycle()	p	p0, p1, … plast, p0, p1, …	cycle('ABCD') --> A B C D A B C D ...
repeat()	elem [,n]	elem, elem, elem, … endlessly or up to n times	repeat(10, 3) --> 10 10 10

Iterators terminating on the shortest input sequence:

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

Combinatoric iterators:

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

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

Itertool functions

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

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

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

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

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

Roughly equivalent to:

def accumulate(iterable, func=operator.add, *, initial=None):
    'Return running totals'
    # accumulate([1,2,3,4,5]) --> 1 3 6 10 15
    # accumulate([1,2,3,4,5], initial=100) --> 100 101 103 106 110 115
    # accumulate([1,2,3,4,5], operator.mul) --> 1 2 6 24 120
    it = iter(iterable)
    total = initial
    if initial is None:
        try:
            total = next(it)
        except StopIteration:
            return
    yield total
    for element in it:
        total = func(total, element)
        yield total

There are a number of uses for the func argument. It can be set to min() for a running minimum, max() for a running maximum, or operator.mul() for a running product. Amortization tables can be built by accumulating interest and applying payments:

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

# Amortize a 5% loan of 1000 with 4 annual payments of 90
>>> cashflows = [1000, -90, -90, -90, -90]
>>> list(accumulate(cashflows, lambda bal, pmt: bal*1.05 + pmt))
[1000, 960.0, 918.0, 873.9000000000001, 827.5950000000001]

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

New in version 3.2.

Changed in version 3.3: Added the optional func parameter.

Changed in version 3.8: Added the optional initial parameter.

itertools.chain(*iterables)

Make an iterator that returns elements from the first iterable until it is exhausted, then proceeds to the next iterable, until all of the iterables are exhausted. Used for treating consecutive sequences as a single sequence. Roughly equivalent to:

def chain(*iterables):
    # chain('ABC', 'DEF') --> A B C D E F
    for it in iterables:
        for element in it:
            yield element

classmethod chain.from_iterable(iterable)

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

def from_iterable(iterables):
    # chain.from_iterable(['ABC', 'DEF']) --> A B C D E F
    for it in iterables:
        for element in it:
            yield element

itertools.combinations(iterable, r)

Return r length subsequences of elements from the input iterable.

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

Elements are treated as unique based on their position, not on their value. So if the input elements are unique, there will be no repeated values in each combination.

Roughly equivalent to:

def combinations(iterable, r):
    # combinations('ABCD', 2) --> AB AC AD BC BD CD
    # combinations(range(4), 3) --> 012 013 023 123
    pool = tuple(iterable)
    n = len(pool)
    if r > n:
        return
    indices = list(range(r))
    yield tuple(pool[i] for i in indices)
    while True:
        for i in reversed(range(r)):
            if indices[i] != i + n - r:
                break
        else:
            return
        indices[i] += 1
        for j in range(i+1, r):
            indices[j] = indices[j-1] + 1
        yield tuple(pool[i] for i in indices)

The code for combinations() can be also expressed as a subsequence of permutations() after filtering entries where the elements are not in sorted order (according to their position in the input pool):

def combinations(iterable, r):
    pool = tuple(iterable)
    n = len(pool)
    for indices in permutations(range(n), r):
        if sorted(indices) == list(indices):
            yield tuple(pool[i] for i in indices)

The number of items returned is n! / r! / (n-r)! when 0 <= r <= n or zero when r > n.

itertools.combinations_with_replacement(iterable, r)

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

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

Elements are treated as unique based on their position, not on their value. So if the input elements are unique, the generated combinations will also be unique.

Roughly equivalent to:

def combinations_with_replacement(iterable, r):
    # combinations_with_replacement('ABC', 2) --> AA AB AC BB BC CC
    pool = tuple(iterable)
    n = len(pool)
    if not n and r:
        return
    indices = [0] * r
    yield tuple(pool[i] for i in indices)
    while True:
        for i in reversed(range(r)):
            if indices[i] != n - 1:
                break
        else:
            return
        indices[i:] = [indices[i] + 1] * (r - i)
        yield tuple(pool[i] for i in indices)

The code for combinations_with_replacement() can be also expressed as a subsequence of product() after filtering entries where the elements are not in sorted order (according to their position in the input pool):

def combinations_with_replacement(iterable, r):
    pool = tuple(iterable)
    n = len(pool)
    for indices in product(range(n), repeat=r):
        if sorted(indices) == list(indices):
            yield tuple(pool[i] for i in indices)

The number of items returned is (n+r-1)! / r! / (n-1)! when n > 0.

New in version 3.1.

itertools.compress(data, selectors)

Make an iterator that filters elements from data returning only those that have a corresponding element in selectors that evaluates to True. Stops when either the data or selectors iterables has been exhausted. Roughly equivalent to:

def compress(data, selectors):
    # compress('ABCDEF', [1,0,1,0,1,1]) --> A C E F
    return (d for d, s in zip(data, selectors) if s)

New in version 3.1.

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

Make an iterator that returns evenly spaced values starting with number start. Often used as an argument to map() to generate consecutive data points. Also, used with zip() to add sequence numbers. Roughly equivalent to:

def count(start=0, step=1):
    # count(10) --> 10 11 12 13 14 ...
    # count(2.5, 0.5) --> 2.5 3.0 3.5 ...
    n = start
    while True:
        yield n
        n += step

When counting with floating point numbers, better accuracy can sometimes be achieved by substituting multiplicative code such as: (start + step * i for i in count()).

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

itertools.cycle(iterable)

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

def cycle(iterable):
    # cycle('ABCD') --> A B C D A B C D A B C D ...
    saved = []
    for element in iterable:
        yield element
        saved.append(element)
    while saved:
        for element in saved:
              yield element

Note, this member of the toolkit may require significant auxiliary storage (depending on the length of the iterable).

itertools.dropwhile(predicate, iterable)

Make an iterator that drops elements from the iterable as long as the predicate is true; afterwards, returns every element. Note, the iterator does not produce any output until the predicate first becomes false, so it may have a lengthy start-up time. Roughly equivalent to:

def dropwhile(predicate, iterable):
    # dropwhile(lambda x: x<5, [1,4,6,4,1]) --> 6 4 1
    iterable = iter(iterable)
    for x in iterable:
        if not predicate(x):
            yield x
            break
    for x in iterable:
        yield x

itertools.filterfalse(predicate, iterable)

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

def filterfalse(predicate, iterable):
    # filterfalse(lambda x: x%2, range(10)) --> 0 2 4 6 8
    if predicate is None:
        predicate = bool
    for x in iterable:
        if not predicate(x):
            yield x

itertools.groupby(iterable, key=None)

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

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

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

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

groupby() is roughly equivalent to:

class groupby:
    # [k for k, g in groupby('AAAABBBCCDAABBB')] --> A B C D A B
    # [list(g) for k, g in groupby('AAAABBBCCD')] --> AAAA BBB CC D

    def __init__(self, iterable, key=None):
        if key is None:
            key = lambda x: x
        self.keyfunc = key
        self.it = iter(iterable)
        self.tgtkey = self.currkey = self.currvalue = object()

    def __iter__(self):
        return self

    def __next__(self):
        self.id = object()
        while self.currkey == self.tgtkey:
            self.currvalue = next(self.it)    # Exit on StopIteration
            self.currkey = self.keyfunc(self.currvalue)
        self.tgtkey = self.currkey
        return (self.currkey, self._grouper(self.tgtkey, self.id))

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

itertools.islice(iterable, stop)

itertools.islice(iterable, start, stop[, step])

Make an iterator that returns selected elements from the iterable. If start is non-zero, then elements from the iterable are skipped until start is reached. Afterward, elements are returned consecutively unless step is set higher than one which results in items being skipped. If stop is None, then iteration continues until the iterator is exhausted, if at all; otherwise, it stops at the specified position.

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

Unlike regular slicing, islice() does not support negative values for start, stop, or step. Can be used to extract related fields from data where the internal structure has been flattened (for example, a multi-line report may list a name field on every third line).

Roughly equivalent to:

def islice(iterable, *args):
    # islice('ABCDEFG', 2) --> A B
    # islice('ABCDEFG', 2, 4) --> C D
    # islice('ABCDEFG', 2, None) --> C D E F G
    # islice('ABCDEFG', 0, None, 2) --> A C E G
    s = slice(*args)
    start, stop, step = s.start or 0, s.stop or sys.maxsize, s.step or 1
    it = iter(range(start, stop, step))
    try:
        nexti = next(it)
    except StopIteration:
        # Consume *iterable* up to the *start* position.
        for i, element in zip(range(start), iterable):
            pass
        return
    try:
        for i, element in enumerate(iterable):
            if i == nexti:
                yield element
                nexti = next(it)
    except StopIteration:
        # Consume to *stop*.
        for i, element in zip(range(i + 1, stop), iterable):
            pass

itertools.pairwise(iterable)

Return successive overlapping pairs taken from the input iterable.

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

Roughly equivalent to:

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

New in version 3.10.

itertools.permutations(iterable, r=None)

Return successive r length permutations of elements in the iterable.

If r is not specified or is None, then r defaults to the length of the iterable and all possible full-length permutations are generated.

The permutation tuples are emitted in lexicographic order according to the order of the input iterable. So, if the input iterable is sorted, the output tuples will be produced in sorted order.

Elements are treated as unique based on their position, not on their value. So if the input elements are unique, there will be no repeated values within a permutation.

Roughly equivalent to:

def permutations(iterable, r=None):
    # permutations('ABCD', 2) --> AB AC AD BA BC BD CA CB CD DA DB DC
    # permutations(range(3)) --> 012 021 102 120 201 210
    pool = tuple(iterable)
    n = len(pool)
    r = n if r is None else r
    if r > n:
        return
    indices = list(range(n))
    cycles = list(range(n, n-r, -1))
    yield tuple(pool[i] for i in indices[:r])
    while n:
        for i in reversed(range(r)):
            cycles[i] -= 1
            if cycles[i] == 0:
                indices[i:] = indices[i+1:] + indices[i:i+1]
                cycles[i] = n - i
            else:
                j = cycles[i]
                indices[i], indices[-j] = indices[-j], indices[i]
                yield tuple(pool[i] for i in indices[:r])
                break
        else:
            return

The code for permutations() can be also expressed as a subsequence of product(), filtered to exclude entries with repeated elements (those from the same position in the input pool):

def permutations(iterable, r=None):
    pool = tuple(iterable)
    n = len(pool)
    r = n if r is None else r
    for indices in product(range(n), repeat=r):
        if len(set(indices)) == r:
            yield tuple(pool[i] for i in indices)

The number of items returned is n! / (n-r)! when 0 <= r <= n or zero when r > n.

itertools.product(*iterables, repeat=1)

Cartesian product of input iterables.

Roughly equivalent to nested for-loops in a generator expression. For example, product(A, B) returns the same as ((x,y) for x in A for y in B).

The nested loops cycle like an odometer with the rightmost element advancing on every iteration. This pattern creates a lexicographic ordering so that if the input’s iterables are sorted, the product tuples are emitted in sorted order.

To compute the product of an iterable with itself, specify the number of repetitions with the optional repeat keyword argument. For example, product(A, repeat=4) means the same as product(A, A, A, A).

This function is roughly equivalent to the following code, except that the actual implementation does not build up intermediate results in memory:

def product(*args, repeat=1):
    # product('ABCD', 'xy') --> Ax Ay Bx By Cx Cy Dx Dy
    # product(range(2), repeat=3) --> 000 001 010 011 100 101 110 111
    pools = [tuple(pool) for pool in args] * repeat
    result = [[]]
    for pool in pools:
        result = [x+[y] for x in result for y in pool]
    for prod in result:
        yield tuple(prod)

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

itertools.repeat(object[, times])

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

Roughly equivalent to:

def repeat(object, times=None):
    # repeat(10, 3) --> 10 10 10
    if times is None:
        while True:
            yield object
    else:
        for i in range(times):
            yield object

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

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

itertools.starmap(function, iterable)

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

The difference between map() and starmap() parallels the distinction between function(a,b) and function(*c). Roughly equivalent to:

def starmap(function, iterable):
    # starmap(pow, [(2,5), (3,2), (10,3)]) --> 32 9 1000
    for args in iterable:
        yield function(*args)

itertools.takewhile(predicate, iterable)

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

def takewhile(predicate, iterable):
    # takewhile(lambda x: x<5, [1,4,6,4,1]) --> 1 4
    for x in iterable:
        if predicate(x):
            yield x
        else:
            break

itertools.tee(iterable, n=2)

Return n independent iterators from a single iterable.

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

def tee(iterable, n=2):
    it = iter(iterable)
    deques = [collections.deque() for i in range(n)]
    def gen(mydeque):
        while True:
            if not mydeque:             # when the local deque is empty
                try:
                    newval = next(it)   # fetch a new value and
                except StopIteration:
                    return
                for d in deques:        # load it to all the deques
                    d.append(newval)
            yield mydeque.popleft()
    return tuple(gen(d) for d in deques)

Once a tee() has been created, the original iterable should not be used anywhere else; otherwise, the iterable could get advanced without the tee objects being informed.

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

This itertool may require significant auxiliary storage (depending on how much temporary data needs to be stored). In general, if one iterator uses most or all of the data before another iterator starts, it is faster to use list() instead of tee().

itertools.zip_longest(*iterables, fillvalue=None)

Make an iterator that aggregates elements from each of the iterables. If the iterables are of uneven length, missing values are filled-in with fillvalue. Iteration continues until the longest iterable is exhausted. Roughly equivalent to:

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

If one of the iterables is potentially infinite, then the zip_longest() function should be wrapped with something that limits the number of calls (for example islice() or takewhile()). If not specified, fillvalue defaults to None.

Itertools Recipes

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

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

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

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

python -m pip install more-itertools

Many of the recipes offer the same high performance as the underlying toolset. Superior memory performance is kept by processing elements one at a time rather than bringing the whole iterable into memory all at once. Code volume is kept small by linking the tools together in a functional style which helps eliminate temporary variables. High speed is retained by preferring “vectorized” building blocks over the use of for-loops and generators which incur interpreter overhead.

import collections
import math
import operator
import random

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

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

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

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

def consume(iterator, n=None):
    "Advance the iterator n-steps ahead. If n is None, consume entirely."
    # Use functions that consume iterators at C speed.
    if n is None:
        # feed the entire iterator into a zero-length deque
        collections.deque(iterator, maxlen=0)
    else:
        # advance to the empty slice starting at position n
        next(islice(iterator, n, n), None)

def nth(iterable, n, default=None):
    "Returns the nth item or a default value"
    return next(islice(iterable, n, None), default)

def all_equal(iterable):
    "Returns True if all the elements are equal to each other"
    g = groupby(iterable)
    return next(g, True) and not next(g, False)

def quantify(iterable, pred=bool):
    "Count how many times the predicate is True"
    return sum(map(pred, iterable))

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

def repeatfunc(func, times=None, *args):
    """Repeat calls to func with specified arguments.

    Example:  repeatfunc(random.random)
    """
    if times is None:
        return starmap(func, repeat(args))
    return starmap(func, repeat(args, times))

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

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

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

def partition(pred, iterable):
    "Use a predicate to partition entries into false entries and true entries"
    # partition(is_odd, range(10)) --> 0 2 4 6 8   and  1 3 5 7 9
    t1, t2 = tee(iterable)
    return filterfalse(pred, t1), filter(pred, t2)

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

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

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

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

def powerset(iterable):
    "powerset([1,2,3]) --> () (1,) (2,) (3,) (1,2) (1,3) (2,3) (1,2,3)"
    s = list(iterable)
    return chain.from_iterable(combinations(s, r) for r in range(len(s)+1))

def unique_everseen(iterable, key=None):
    "List unique elements, preserving order. Remember all elements ever seen."
    # unique_everseen('AAAABBBCCDAABBB') --> A B C D
    # unique_everseen('ABBcCAD', str.lower) --> A B c D
    seen = set()
    if key is None:
        for element in filterfalse(seen.__contains__, iterable):
            seen.add(element)
            yield element
        # For order preserving deduplication,
        # a faster but non-lazy solution is:
        #     yield from dict.fromkeys(iterable)
    else:
        for element in iterable:
            k = key(element)
            if k not in seen:
                seen.add(k)
                yield element
        # For use cases that allow the last matching element to be returned,
        # a faster but non-lazy solution is:
        #      t1, t2 = tee(iterable)
        #      yield from dict(zip(map(key, t1), t2)).values()

def unique_justseen(iterable, key=None):
    "List unique elements, preserving order. Remember only the element just seen."
    # unique_justseen('AAAABBBCCDAABBB') --> A B C D A B
    # unique_justseen('ABBcCAD', str.lower) --> A B c A D
    return map(next, map(operator.itemgetter(1), groupby(iterable, key)))

def iter_except(func, exception, first=None):
    """ Call a function repeatedly until an exception is raised.

    Converts a call-until-exception interface to an iterator interface.
    Like builtins.iter(func, sentinel) but uses an exception instead
    of a sentinel to end the loop.

    Examples:
        iter_except(functools.partial(heappop, h), IndexError)   # priority queue iterator
        iter_except(d.popitem, KeyError)                         # non-blocking dict iterator
        iter_except(d.popleft, IndexError)                       # non-blocking deque iterator
        iter_except(q.get_nowait, Queue.Empty)                   # loop over a producer Queue
        iter_except(s.pop, KeyError)                             # non-blocking set iterator

    """
    try:
        if first is not None:
            yield first()            # For database APIs needing an initial cast to db.first()
        while True:
            yield func()
    except exception:
        pass

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

    If no true value is found, returns *default*

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

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

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