1476 lines
41 KiB
Python
1476 lines
41 KiB
Python
"""Imported from the recipes section of the itertools documentation.
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All functions taken from the recipes section of the itertools library docs
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[1]_.
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Some backward-compatible usability improvements have been made.
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.. [1] http://docs.python.org/library/itertools.html#recipes
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"""
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import random
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from bisect import bisect_left, insort
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from collections import deque
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from contextlib import suppress
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from functools import lru_cache, reduce
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from heapq import heappush, heappushpop
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from itertools import (
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accumulate,
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chain,
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combinations,
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compress,
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count,
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cycle,
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filterfalse,
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groupby,
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islice,
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pairwise as itertools_pairwise,
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product,
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repeat,
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starmap,
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takewhile,
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tee,
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zip_longest,
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)
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from math import prod, comb, isqrt, gcd
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from operator import mul, getitem, index, is_, itemgetter, truediv
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from random import randrange, sample, choice, shuffle
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from sys import hexversion
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__all__ = [
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'all_equal',
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'batched',
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'before_and_after',
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'consume',
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'convolve',
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'dotproduct',
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'first_true',
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'factor',
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'flatten',
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'grouper',
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'is_prime',
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'iter_except',
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'iter_index',
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'loops',
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'matmul',
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'multinomial',
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'ncycles',
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'nth',
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'nth_combination',
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'padnone',
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'pad_none',
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'pairwise',
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'partition',
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'polynomial_eval',
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'polynomial_from_roots',
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'polynomial_derivative',
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'powerset',
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'prepend',
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'quantify',
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'reshape',
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'random_combination_with_replacement',
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'random_combination',
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'random_derangement',
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'random_permutation',
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'random_product',
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'repeatfunc',
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'roundrobin',
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'running_mean',
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'running_median',
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'sieve',
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'sliding_window',
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'subslices',
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'sum_of_squares',
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'tabulate',
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'tail',
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'take',
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'totient',
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'transpose',
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'triplewise',
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'unique',
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'unique_everseen',
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'unique_justseen',
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]
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_marker = object()
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# heapq max-heap functions are available for Python 3.14+
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try:
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from heapq import heappush_max, heappushpop_max
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except ImportError: # pragma: no cover
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_max_heap_available = False
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else: # pragma: no cover
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_max_heap_available = True
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def take(n, iterable):
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"""Return first *n* items of the *iterable* as a list.
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>>> take(3, range(10))
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[0, 1, 2]
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If there are fewer than *n* items in the iterable, all of them are
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returned.
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>>> take(10, range(3))
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[0, 1, 2]
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"""
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return list(islice(iterable, n))
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def tabulate(function, start=0):
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"""Return an iterator over the results of ``func(start)``,
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``func(start + 1)``, ``func(start + 2)``...
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*func* should be a function that accepts one integer argument.
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If *start* is not specified it defaults to 0. It will be incremented each
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time the iterator is advanced.
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>>> square = lambda x: x ** 2
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>>> iterator = tabulate(square, -3)
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>>> take(4, iterator)
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[9, 4, 1, 0]
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"""
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return map(function, count(start))
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def tail(n, iterable):
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"""Return an iterator over the last *n* items of *iterable*.
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>>> t = tail(3, 'ABCDEFG')
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>>> list(t)
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['E', 'F', 'G']
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"""
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try:
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size = len(iterable)
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except TypeError:
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return iter(deque(iterable, maxlen=n))
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else:
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return islice(iterable, max(0, size - n), None)
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def consume(iterator, n=None):
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"""Advance *iterable* by *n* steps. If *n* is ``None``, consume it
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entirely.
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Efficiently exhausts an iterator without returning values. Defaults to
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consuming the whole iterator, but an optional second argument may be
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provided to limit consumption.
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>>> i = (x for x in range(10))
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>>> next(i)
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0
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>>> consume(i, 3)
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>>> next(i)
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4
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>>> consume(i)
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>>> next(i)
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Traceback (most recent call last):
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File "<stdin>", line 1, in <module>
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StopIteration
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If the iterator has fewer items remaining than the provided limit, the
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whole iterator will be consumed.
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>>> i = (x for x in range(3))
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>>> consume(i, 5)
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>>> next(i)
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Traceback (most recent call last):
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File "<stdin>", line 1, in <module>
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StopIteration
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"""
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# Use functions that consume iterators at C speed.
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if n is None:
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# feed the entire iterator into a zero-length deque
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deque(iterator, maxlen=0)
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else:
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# advance to the empty slice starting at position n
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next(islice(iterator, n, n), None)
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def nth(iterable, n, default=None):
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"""Returns the nth item or a default value.
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>>> l = range(10)
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>>> nth(l, 3)
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3
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>>> nth(l, 20, "zebra")
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'zebra'
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"""
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return next(islice(iterable, n, None), default)
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def all_equal(iterable, key=None):
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"""
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Returns ``True`` if all the elements are equal to each other.
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>>> all_equal('aaaa')
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True
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>>> all_equal('aaab')
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False
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A function that accepts a single argument and returns a transformed version
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of each input item can be specified with *key*:
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>>> all_equal('AaaA', key=str.casefold)
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True
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>>> all_equal([1, 2, 3], key=lambda x: x < 10)
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True
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"""
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iterator = groupby(iterable, key)
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for first in iterator:
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for second in iterator:
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return False
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return True
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return True
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def quantify(iterable, pred=bool):
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"""Return the how many times the predicate is true.
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>>> quantify([True, False, True])
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2
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"""
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return sum(map(pred, iterable))
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def pad_none(iterable):
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"""Returns the sequence of elements and then returns ``None`` indefinitely.
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>>> take(5, pad_none(range(3)))
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[0, 1, 2, None, None]
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Useful for emulating the behavior of the built-in :func:`map` function.
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See also :func:`padded`.
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"""
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return chain(iterable, repeat(None))
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padnone = pad_none
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def ncycles(iterable, n):
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"""Returns the sequence elements *n* times
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>>> list(ncycles(["a", "b"], 3))
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['a', 'b', 'a', 'b', 'a', 'b']
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"""
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return chain.from_iterable(repeat(tuple(iterable), n))
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def dotproduct(vec1, vec2):
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"""Returns the dot product of the two iterables.
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>>> dotproduct([10, 15, 12], [0.65, 0.80, 1.25])
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33.5
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>>> 10 * 0.65 + 15 * 0.80 + 12 * 1.25
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33.5
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In Python 3.12 and later, use ``math.sumprod()`` instead.
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"""
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return sum(map(mul, vec1, vec2))
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# math.sumprod is available for Python 3.12+
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try:
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from math import sumprod as _sumprod
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except ImportError: # pragma: no cover
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_sumprod = dotproduct
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def flatten(list_of_lists):
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"""Return an iterator flattening one level of nesting in a list of lists.
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>>> list(flatten([[0, 1], [2, 3]]))
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[0, 1, 2, 3]
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See also :func:`collapse`, which can flatten multiple levels of nesting.
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"""
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return chain.from_iterable(list_of_lists)
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def repeatfunc(function, times=None, *args):
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"""Call *function* with *args* repeatedly, returning an iterable over the
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results.
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If *times* is specified, the iterable will terminate after that many
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repetitions:
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>>> from operator import add
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>>> times = 4
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>>> args = 3, 5
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>>> list(repeatfunc(add, times, *args))
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[8, 8, 8, 8]
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If *times* is ``None`` the iterable will not terminate:
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>>> from random import randrange
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>>> times = None
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>>> args = 1, 11
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>>> take(6, repeatfunc(randrange, times, *args)) # doctest:+SKIP
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[2, 4, 8, 1, 8, 4]
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"""
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if times is None:
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return starmap(function, repeat(args))
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return starmap(function, repeat(args, times))
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def pairwise(iterable):
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"""
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Wrapper for :func:`itertools.pairwise`.
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.. warning::
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This function is deprecated as of version 11.0.0. It will be removed in a future
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major release.
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"""
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return itertools_pairwise(iterable)
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def grouper(iterable, n, incomplete='fill', fillvalue=None):
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"""Group elements from *iterable* into fixed-length groups of length *n*.
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>>> list(grouper('ABCDEF', 3))
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[('A', 'B', 'C'), ('D', 'E', 'F')]
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The keyword arguments *incomplete* and *fillvalue* control what happens for
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iterables whose length is not a multiple of *n*.
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When *incomplete* is `'fill'`, the last group will contain instances of
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*fillvalue*.
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>>> list(grouper('ABCDEFG', 3, incomplete='fill', fillvalue='x'))
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[('A', 'B', 'C'), ('D', 'E', 'F'), ('G', 'x', 'x')]
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When *incomplete* is `'ignore'`, the last group will not be emitted.
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>>> list(grouper('ABCDEFG', 3, incomplete='ignore', fillvalue='x'))
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[('A', 'B', 'C'), ('D', 'E', 'F')]
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When *incomplete* is `'strict'`, a `ValueError` will be raised.
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>>> iterator = grouper('ABCDEFG', 3, incomplete='strict')
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>>> list(iterator) # doctest: +IGNORE_EXCEPTION_DETAIL
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Traceback (most recent call last):
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...
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ValueError
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"""
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iterators = [iter(iterable)] * n
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match incomplete:
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case 'fill':
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return zip_longest(*iterators, fillvalue=fillvalue)
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case 'strict':
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return zip(*iterators, strict=True)
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case 'ignore':
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return zip(*iterators)
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case _:
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raise ValueError('Expected fill, strict, or ignore')
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def roundrobin(*iterables):
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"""Visit input iterables in a cycle until each is exhausted.
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>>> list(roundrobin('ABC', 'D', 'EF'))
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['A', 'D', 'E', 'B', 'F', 'C']
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This function produces the same output as :func:`interleave_longest`, but
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may perform better for some inputs (in particular when the number of
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iterables is small).
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"""
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# Algorithm credited to George Sakkis
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iterators = map(iter, iterables)
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for num_active in range(len(iterables), 0, -1):
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iterators = cycle(islice(iterators, num_active))
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yield from map(next, iterators)
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def partition(pred, iterable):
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"""
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Returns a 2-tuple of iterables derived from the input iterable.
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The first yields the items that have ``pred(item) == False``.
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The second yields the items that have ``pred(item) == True``.
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>>> is_odd = lambda x: x % 2 != 0
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>>> iterable = range(10)
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>>> even_items, odd_items = partition(is_odd, iterable)
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>>> list(even_items), list(odd_items)
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([0, 2, 4, 6, 8], [1, 3, 5, 7, 9])
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If *pred* is None, :func:`bool` is used.
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>>> iterable = [0, 1, False, True, '', ' ']
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>>> false_items, true_items = partition(None, iterable)
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>>> list(false_items), list(true_items)
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([0, False, ''], [1, True, ' '])
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"""
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if pred is None:
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pred = bool
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iterator = iter(iterable)
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false_queue = deque()
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true_queue = deque()
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def gen(queue):
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while True:
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while queue:
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yield queue.popleft()
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for value in iterator:
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(true_queue if pred(value) else false_queue).append(value)
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break
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else:
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return
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return gen(false_queue), gen(true_queue)
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def powerset(iterable):
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"""Yields all possible subsets of the iterable.
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>>> list(powerset([1, 2, 3]))
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[(), (1,), (2,), (3,), (1, 2), (1, 3), (2, 3), (1, 2, 3)]
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:func:`powerset` will operate on iterables that aren't :class:`set`
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instances, so repeated elements in the input will produce repeated elements
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in the output.
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>>> seq = [1, 1, 0]
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>>> list(powerset(seq))
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[(), (1,), (1,), (0,), (1, 1), (1, 0), (1, 0), (1, 1, 0)]
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For a variant that efficiently yields actual :class:`set` instances, see
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:func:`powerset_of_sets`.
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"""
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s = list(iterable)
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return chain.from_iterable(combinations(s, r) for r in range(len(s) + 1))
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def unique_everseen(iterable, key=None):
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"""Yield unique elements, preserving order. Remember all elements ever seen.
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||
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>>> list(unique_everseen('AAAABBBCCDAABBB'))
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['A', 'B', 'C', 'D']
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>>> list(unique_everseen('ABBCcAD', str.casefold))
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||
['A', 'B', 'C', 'D']
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||
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||
Raises ``TypeError`` for unhashable items.
|
||
|
||
Some unhashable objects can be converted to hashable objects
|
||
using the *key* parameter:
|
||
|
||
* For ``list`` objects, try ``key=tuple``.
|
||
* For ``set`` objects, try ``key=frozenset``.
|
||
* For ``dict`` objects, try ``key=lambda x: frozenset(x.items())``
|
||
or in Python 3.15 and later, set ``key=frozendict``.
|
||
|
||
Alternatively, consider the ``unique()`` itertool recipe. It sorts
|
||
the data and then uses equality to eliminate duplicates. Hashability
|
||
is not required.
|
||
|
||
"""
|
||
seen = set()
|
||
if key is None:
|
||
for element in filterfalse(seen.__contains__, iterable):
|
||
seen.add(element)
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||
yield element
|
||
else:
|
||
for element in iterable:
|
||
k = key(element)
|
||
if k not in seen:
|
||
seen.add(k)
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||
yield element
|
||
|
||
|
||
def unique_justseen(iterable, key=None):
|
||
"""Yields elements in order, ignoring serial duplicates
|
||
|
||
>>> list(unique_justseen('AAAABBBCCDAABBB'))
|
||
['A', 'B', 'C', 'D', 'A', 'B']
|
||
>>> list(unique_justseen('ABBCcAD', str.lower))
|
||
['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(iterable, key=None, reverse=False):
|
||
"""Yields unique elements in sorted order.
|
||
|
||
>>> list(unique([[1, 2], [3, 4], [1, 2]]))
|
||
[[1, 2], [3, 4]]
|
||
|
||
*key* and *reverse* are passed to :func:`sorted`.
|
||
|
||
>>> list(unique('ABBcCAD', str.casefold))
|
||
['A', 'B', 'c', 'D']
|
||
>>> list(unique('ABBcCAD', str.casefold, reverse=True))
|
||
['D', 'c', 'B', 'A']
|
||
|
||
The elements in *iterable* need not be hashable, but they must be
|
||
comparable for sorting to work.
|
||
"""
|
||
sequenced = sorted(iterable, key=key, reverse=reverse)
|
||
return unique_justseen(sequenced, key=key)
|
||
|
||
|
||
def iter_except(function, exception, first=None):
|
||
"""Yields results from a function repeatedly until an exception is raised.
|
||
|
||
Converts a call-until-exception interface to an iterator interface.
|
||
Like ``iter(function, sentinel)``, but uses an exception instead of a sentinel
|
||
to end the loop.
|
||
|
||
>>> l = [0, 1, 2]
|
||
>>> list(iter_except(l.pop, IndexError))
|
||
[2, 1, 0]
|
||
|
||
Multiple exceptions can be specified as a stopping condition:
|
||
|
||
>>> l = [1, 2, 3, '...', 4, 5, 6]
|
||
>>> list(iter_except(lambda: 1 + l.pop(), (IndexError, TypeError)))
|
||
[7, 6, 5]
|
||
>>> list(iter_except(lambda: 1 + l.pop(), (IndexError, TypeError)))
|
||
[4, 3, 2]
|
||
>>> list(iter_except(lambda: 1 + l.pop(), (IndexError, TypeError)))
|
||
[]
|
||
|
||
"""
|
||
with suppress(exception):
|
||
if first is not None:
|
||
yield first()
|
||
while True:
|
||
yield function()
|
||
|
||
|
||
def first_true(iterable, default=None, 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) == True`` .
|
||
|
||
>>> first_true(range(10))
|
||
1
|
||
>>> first_true(range(10), pred=lambda x: x > 5)
|
||
6
|
||
>>> first_true(range(10), default='missing', pred=lambda x: x > 9)
|
||
'missing'
|
||
|
||
"""
|
||
return next(filter(pred, iterable), default)
|
||
|
||
|
||
def random_product(*iterables, repeat=1):
|
||
"""Draw an item at random from each of the input iterables.
|
||
|
||
>>> random_product('abc', range(4), 'XYZ') # doctest:+SKIP
|
||
('c', 3, 'Z')
|
||
|
||
If *repeat* is provided as a keyword argument, that many items will be
|
||
drawn from each iterable.
|
||
|
||
>>> random_product('abcd', range(4), repeat=2) # doctest:+SKIP
|
||
('a', 2, 'd', 3)
|
||
|
||
This equivalent to taking a random selection from
|
||
``itertools.product(*args, repeat=repeat)``.
|
||
|
||
"""
|
||
pools = tuple(map(tuple, iterables)) * repeat
|
||
return tuple(map(choice, pools))
|
||
|
||
|
||
def random_permutation(iterable, r=None):
|
||
"""Return a random *r* length permutation of the elements in *iterable*.
|
||
|
||
If *r* is not specified or is ``None``, then *r* defaults to the length of
|
||
*iterable*.
|
||
|
||
>>> random_permutation(range(5)) # doctest:+SKIP
|
||
(3, 4, 0, 1, 2)
|
||
|
||
This equivalent to taking a random selection from
|
||
``itertools.permutations(iterable, r)``.
|
||
|
||
"""
|
||
pool = tuple(iterable)
|
||
r = len(pool) if r is None else r
|
||
return tuple(sample(pool, r))
|
||
|
||
|
||
def random_combination(iterable, r):
|
||
"""Return a random *r* length subsequence of the elements in *iterable*.
|
||
|
||
>>> random_combination(range(5), 3) # doctest:+SKIP
|
||
(2, 3, 4)
|
||
|
||
This equivalent to taking a random selection from
|
||
``itertools.combinations(iterable, r)``.
|
||
|
||
"""
|
||
pool = tuple(iterable)
|
||
n = len(pool)
|
||
indices = sorted(sample(range(n), r))
|
||
return tuple([pool[i] for i in indices])
|
||
|
||
|
||
def random_combination_with_replacement(iterable, r):
|
||
"""Return a random *r* length subsequence of elements in *iterable*,
|
||
allowing individual elements to be repeated.
|
||
|
||
>>> random_combination_with_replacement(range(3), 5) # doctest:+SKIP
|
||
(0, 0, 1, 2, 2)
|
||
|
||
This equivalent to taking a random selection from
|
||
``itertools.combinations_with_replacement(iterable, r)``.
|
||
|
||
"""
|
||
pool = tuple(iterable)
|
||
n = len(pool)
|
||
indices = sorted(randrange(n) for i in range(r))
|
||
return tuple([pool[i] for i in indices])
|
||
|
||
|
||
def nth_combination(iterable, r, index):
|
||
"""Equivalent to ``list(combinations(iterable, r))[index]``.
|
||
|
||
The subsequences of *iterable* that are of length *r* can be ordered
|
||
lexicographically. :func:`nth_combination` computes the subsequence at
|
||
sort position *index* directly, without computing the previous
|
||
subsequences.
|
||
|
||
>>> nth_combination(range(5), 3, 5)
|
||
(0, 3, 4)
|
||
|
||
``ValueError`` will be raised If *r* is negative.
|
||
``IndexError`` will be raised if the given *index* is invalid.
|
||
"""
|
||
pool = tuple(iterable)
|
||
n = len(pool)
|
||
c = comb(n, r)
|
||
|
||
if index < 0:
|
||
index += c
|
||
if not 0 <= 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 prepend(value, iterable):
|
||
"""Yield *value*, followed by the elements in *iterable*.
|
||
|
||
>>> value = '0'
|
||
>>> iterable = ['1', '2', '3']
|
||
>>> list(prepend(value, iterable))
|
||
['0', '1', '2', '3']
|
||
|
||
To prepend multiple values, see :func:`itertools.chain`
|
||
or :func:`value_chain`.
|
||
|
||
"""
|
||
return chain([value], iterable)
|
||
|
||
|
||
def convolve(signal, kernel):
|
||
"""Discrete linear convolution of two iterables.
|
||
Equivalent to polynomial multiplication.
|
||
|
||
For example, multiplying ``(x² -x - 20)`` by ``(x - 3)``
|
||
gives ``(x³ -4x² -17x + 60)``.
|
||
|
||
>>> list(convolve([1, -1, -20], [1, -3]))
|
||
[1, -4, -17, 60]
|
||
|
||
Examples of popular kinds of kernels:
|
||
|
||
* The kernel ``[0.25, 0.25, 0.25, 0.25]`` computes a moving average.
|
||
For image data, this blurs the image and reduces noise.
|
||
* The kernel ``[1/2, 0, -1/2]`` estimates the first derivative of
|
||
a function evaluated at evenly spaced inputs.
|
||
* The kernel ``[1, -2, 1]`` estimates the second derivative of a
|
||
function evaluated at evenly spaced inputs.
|
||
|
||
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.
|
||
|
||
Supports all numeric types: int, float, complex, Decimal, Fraction.
|
||
|
||
References:
|
||
|
||
* Article: https://betterexplained.com/articles/intuitive-convolution/
|
||
* Video by 3Blue1Brown: https://www.youtube.com/watch?v=KuXjwB4LzSA
|
||
|
||
"""
|
||
# This implementation comes from an older version of the itertools
|
||
# documentation. While the newer implementation is a bit clearer,
|
||
# this one was kept because the inlined window logic is faster
|
||
# and it avoids an unnecessary deque-to-tuple conversion.
|
||
kernel = tuple(kernel)[::-1]
|
||
n = len(kernel)
|
||
window = deque([0], maxlen=n) * n
|
||
for x in chain(signal, repeat(0, n - 1)):
|
||
window.append(x)
|
||
yield _sumprod(kernel, window)
|
||
|
||
|
||
def before_and_after(predicate, it):
|
||
"""A variant of :func:`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.
|
||
"""
|
||
trues, after = tee(it)
|
||
trues = compress(takewhile(predicate, trues), zip(after))
|
||
return trues, after
|
||
|
||
|
||
def triplewise(iterable):
|
||
"""Return overlapping triplets from *iterable*.
|
||
|
||
>>> list(triplewise('ABCDE'))
|
||
[('A', 'B', 'C'), ('B', 'C', 'D'), ('C', 'D', 'E')]
|
||
|
||
"""
|
||
# This deviates from the itertools documentation recipe - see
|
||
# https://github.com/more-itertools/more-itertools/issues/889
|
||
t1, t2, t3 = tee(iterable, 3)
|
||
next(t3, None)
|
||
next(t3, None)
|
||
next(t2, None)
|
||
return zip(t1, t2, t3)
|
||
|
||
|
||
def _sliding_window_islice(iterable, n):
|
||
# Fast path for small, non-zero values of n.
|
||
iterators = tee(iterable, n)
|
||
for i, iterator in enumerate(iterators):
|
||
next(islice(iterator, i, i), None)
|
||
return zip(*iterators)
|
||
|
||
|
||
def _sliding_window_deque(iterable, n):
|
||
# Normal path for other values of n.
|
||
iterator = iter(iterable)
|
||
window = deque(islice(iterator, n - 1), maxlen=n)
|
||
for x in iterator:
|
||
window.append(x)
|
||
yield tuple(window)
|
||
|
||
|
||
def sliding_window(iterable, n):
|
||
"""Return a sliding window of width *n* over *iterable*.
|
||
|
||
>>> list(sliding_window(range(6), 4))
|
||
[(0, 1, 2, 3), (1, 2, 3, 4), (2, 3, 4, 5)]
|
||
|
||
If *iterable* has fewer than *n* items, then nothing is yielded:
|
||
|
||
>>> list(sliding_window(range(3), 4))
|
||
[]
|
||
|
||
For a variant with more features, see :func:`windowed`.
|
||
"""
|
||
if n > 20:
|
||
return _sliding_window_deque(iterable, n)
|
||
elif n > 2:
|
||
return _sliding_window_islice(iterable, n)
|
||
elif n == 2:
|
||
return pairwise(iterable)
|
||
elif n == 1:
|
||
return zip(iterable)
|
||
else:
|
||
raise ValueError(f'n should be at least one, not {n}')
|
||
|
||
|
||
def subslices(iterable):
|
||
"""Return all contiguous non-empty subslices of *iterable*.
|
||
|
||
>>> list(subslices('ABC'))
|
||
[['A'], ['A', 'B'], ['A', 'B', 'C'], ['B'], ['B', 'C'], ['C']]
|
||
|
||
This is similar to :func:`substrings`, but emits items in a different
|
||
order.
|
||
"""
|
||
seq = list(iterable)
|
||
slices = starmap(slice, combinations(range(len(seq) + 1), 2))
|
||
return map(getitem, repeat(seq), slices)
|
||
|
||
|
||
def polynomial_from_roots(roots):
|
||
"""Compute a polynomial's coefficients from its roots.
|
||
|
||
>>> roots = [5, -4, 3] # (x - 5) * (x + 4) * (x - 3)
|
||
>>> polynomial_from_roots(roots) # x³ - 4 x² - 17 x + 60
|
||
[1, -4, -17, 60]
|
||
|
||
Note that polynomial coefficients are specified in descending power order.
|
||
|
||
Supports all numeric types: int, float, complex, Decimal, Fraction.
|
||
"""
|
||
|
||
# This recipe differs from the one in itertools docs in that it
|
||
# applies list() after each call to convolve(). This avoids
|
||
# hitting stack limits with nested generators.
|
||
|
||
poly = [1]
|
||
for root in roots:
|
||
poly = list(convolve(poly, (1, -root)))
|
||
return poly
|
||
|
||
|
||
def iter_index(iterable, value, start=0, stop=None):
|
||
"""Yield the index of each place in *iterable* that *value* occurs,
|
||
beginning with index *start* and ending before index *stop*.
|
||
|
||
|
||
>>> list(iter_index('AABCADEAF', 'A'))
|
||
[0, 1, 4, 7]
|
||
>>> list(iter_index('AABCADEAF', 'A', 1)) # start index is inclusive
|
||
[1, 4, 7]
|
||
>>> list(iter_index('AABCADEAF', 'A', 1, 7)) # stop index is not inclusive
|
||
[1, 4]
|
||
|
||
The behavior for non-scalar *values* matches the built-in Python types.
|
||
|
||
>>> list(iter_index('ABCDABCD', 'AB'))
|
||
[0, 4]
|
||
>>> list(iter_index([0, 1, 2, 3, 0, 1, 2, 3], [0, 1]))
|
||
[]
|
||
>>> list(iter_index([[0, 1], [2, 3], [0, 1], [2, 3]], [0, 1]))
|
||
[0, 2]
|
||
|
||
See :func:`locate` for a more general means of finding the indexes
|
||
associated with particular values.
|
||
|
||
"""
|
||
seq_index = getattr(iterable, 'index', None)
|
||
if seq_index is None:
|
||
# Slow path for general iterables
|
||
iterator = islice(iterable, start, stop)
|
||
for i, element in enumerate(iterator, start):
|
||
if element is value or element == value:
|
||
yield i
|
||
else:
|
||
# Fast path for sequences
|
||
stop = len(iterable) if stop is None else stop
|
||
i = start - 1
|
||
with suppress(ValueError):
|
||
while True:
|
||
yield (i := seq_index(value, i + 1, stop))
|
||
|
||
|
||
def sieve(n):
|
||
"""Yield the primes less than n.
|
||
|
||
>>> list(sieve(30))
|
||
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
|
||
|
||
"""
|
||
# This implementation comes from an older version of the itertools
|
||
# documentation. The newer implementation is easier to read but is
|
||
# less lazy.
|
||
if n > 2:
|
||
yield 2
|
||
start = 3
|
||
data = bytearray((0, 1)) * (n // 2)
|
||
for p in iter_index(data, 1, start, stop=isqrt(n) + 1):
|
||
yield from iter_index(data, 1, start, p * p)
|
||
data[p * p : n : p + p] = bytes(len(range(p * p, n, p + p)))
|
||
start = p * p
|
||
yield from iter_index(data, 1, start)
|
||
|
||
|
||
def _batched(iterable, n, *, strict=False): # pragma: no cover
|
||
"""Batch data into tuples of length *n*. If the number of items in
|
||
*iterable* is not divisible by *n*:
|
||
* The last batch will be shorter if *strict* is ``False``.
|
||
* :exc:`ValueError` will be raised if *strict* is ``True``.
|
||
|
||
>>> list(batched('ABCDEFG', 3))
|
||
[('A', 'B', 'C'), ('D', 'E', 'F'), ('G',)]
|
||
|
||
On Python 3.13 and above, this is an alias for :func:`itertools.batched`.
|
||
"""
|
||
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
|
||
|
||
|
||
if hexversion >= 0x30D00A2: # pragma: no cover
|
||
from itertools import batched as itertools_batched
|
||
|
||
def batched(iterable, n, *, strict=False):
|
||
return itertools_batched(iterable, n, strict=strict)
|
||
|
||
batched.__doc__ = _batched.__doc__
|
||
else: # pragma: no cover
|
||
batched = _batched
|
||
|
||
|
||
def transpose(matrix):
|
||
"""Swap the rows and columns of the input matrix.
|
||
|
||
>>> list(transpose([(1, 2, 3), (11, 22, 33)]))
|
||
[(1, 11), (2, 22), (3, 33)]
|
||
|
||
The caller should ensure that the dimensions of the input are compatible.
|
||
If the input is empty, no output will be produced.
|
||
"""
|
||
return zip(*matrix, strict=True)
|
||
|
||
|
||
def _is_scalar(value, stringlike=(str, bytes)):
|
||
"Scalars are bytes, strings, and non-iterables."
|
||
try:
|
||
iter(value)
|
||
except TypeError:
|
||
return True
|
||
return isinstance(value, stringlike)
|
||
|
||
|
||
def _flatten_tensor(tensor):
|
||
"Depth-first iterator over scalars in a tensor."
|
||
iterator = iter(tensor)
|
||
while True:
|
||
try:
|
||
value = next(iterator)
|
||
except StopIteration:
|
||
return iterator
|
||
iterator = chain((value,), iterator)
|
||
if _is_scalar(value):
|
||
return iterator
|
||
iterator = chain.from_iterable(iterator)
|
||
|
||
|
||
def reshape(matrix, shape):
|
||
"""Change the shape of a *matrix*.
|
||
|
||
If *shape* is an integer, the matrix must be two dimensional
|
||
and the shape is interpreted as the desired number of columns:
|
||
|
||
>>> matrix = [(0, 1), (2, 3), (4, 5)]
|
||
>>> cols = 3
|
||
>>> list(reshape(matrix, cols))
|
||
[(0, 1, 2), (3, 4, 5)]
|
||
|
||
If *shape* is a tuple (or other iterable), the input matrix can have
|
||
any number of dimensions. It will first be flattened and then rebuilt
|
||
to the desired shape which can also be multidimensional:
|
||
|
||
>>> matrix = [(0, 1), (2, 3), (4, 5)] # Start with a 3 x 2 matrix
|
||
|
||
>>> list(reshape(matrix, (2, 3))) # Make a 2 x 3 matrix
|
||
[(0, 1, 2), (3, 4, 5)]
|
||
|
||
>>> list(reshape(matrix, (6,))) # Make a vector of length six
|
||
[0, 1, 2, 3, 4, 5]
|
||
|
||
>>> list(reshape(matrix, (2, 1, 3, 1))) # Make 2 x 1 x 3 x 1 tensor
|
||
[(((0,), (1,), (2,)),), (((3,), (4,), (5,)),)]
|
||
|
||
Each dimension is assumed to be uniform, either all arrays or all scalars.
|
||
Flattening stops when the first value in a dimension is a scalar.
|
||
Scalars are bytes, strings, and non-iterables.
|
||
The reshape iterator stops when the requested shape is complete
|
||
or when the input is exhausted, whichever comes first.
|
||
|
||
"""
|
||
if isinstance(shape, int):
|
||
return batched(chain.from_iterable(matrix), shape)
|
||
first_dim, *dims = shape
|
||
scalar_stream = _flatten_tensor(matrix)
|
||
reshaped = reduce(batched, reversed(dims), scalar_stream)
|
||
return islice(reshaped, first_dim)
|
||
|
||
|
||
def matmul(m1, m2):
|
||
"""Multiply two matrices.
|
||
|
||
>>> list(matmul([(7, 5), (3, 5)], [(2, 5), (7, 9)]))
|
||
[(49, 80), (41, 60)]
|
||
|
||
The caller should ensure that the dimensions of the input matrices are
|
||
compatible with each other.
|
||
|
||
Supports all numeric types: int, float, complex, Decimal, Fraction.
|
||
"""
|
||
n = len(m2[0])
|
||
return batched(starmap(_sumprod, product(m1, transpose(m2))), n)
|
||
|
||
|
||
def _factor_pollard(n):
|
||
# Return a factor of n using Pollard's rho algorithm.
|
||
# Efficient when n is odd and composite.
|
||
for b in range(1, n):
|
||
x = y = 2
|
||
d = 1
|
||
while d == 1:
|
||
x = (x * x + b) % n
|
||
y = (y * y + b) % n
|
||
y = (y * y + b) % n
|
||
d = gcd(x - y, n)
|
||
if d != n:
|
||
return d
|
||
raise ValueError('prime or under 5') # pragma: no cover
|
||
|
||
|
||
_primes_below_211 = tuple(sieve(211))
|
||
|
||
|
||
def factor(n):
|
||
"""Yield the prime factors of n.
|
||
|
||
>>> list(factor(360))
|
||
[2, 2, 2, 3, 3, 5]
|
||
|
||
Finds small factors with trial division. Larger factors are
|
||
either verified as prime with ``is_prime`` or split into
|
||
smaller factors with Pollard's rho algorithm.
|
||
"""
|
||
|
||
# Corner case reduction
|
||
if n < 2:
|
||
return
|
||
|
||
# Trial division reduction
|
||
for prime in _primes_below_211:
|
||
while not n % prime:
|
||
yield prime
|
||
n //= prime
|
||
|
||
# Pollard's rho reduction
|
||
primes = []
|
||
todo = [n] if n > 1 else []
|
||
for n in todo:
|
||
if n < 211**2 or is_prime(n):
|
||
primes.append(n)
|
||
else:
|
||
fact = _factor_pollard(n)
|
||
todo += (fact, n // fact)
|
||
yield from sorted(primes)
|
||
|
||
|
||
def polynomial_eval(coefficients, x):
|
||
"""Evaluate a polynomial at a specific value.
|
||
|
||
Computes with better numeric stability than Horner's method.
|
||
|
||
Evaluate ``x^3 - 4 * x^2 - 17 * x + 60`` at ``x = 2.5``:
|
||
|
||
>>> coefficients = [1, -4, -17, 60]
|
||
>>> x = 2.5
|
||
>>> polynomial_eval(coefficients, x)
|
||
8.125
|
||
|
||
Note that polynomial coefficients are specified in descending power order.
|
||
|
||
Supports all numeric types: int, float, complex, Decimal, Fraction.
|
||
"""
|
||
n = len(coefficients)
|
||
if n == 0:
|
||
return type(x)(0)
|
||
powers = map(pow, repeat(x), reversed(range(n)))
|
||
return _sumprod(coefficients, powers)
|
||
|
||
|
||
def sum_of_squares(iterable):
|
||
"""Return the sum of the squares of the input values.
|
||
|
||
>>> sum_of_squares([10, 20, 30])
|
||
1400
|
||
|
||
Supports all numeric types: int, float, complex, Decimal, Fraction.
|
||
"""
|
||
return _sumprod(*tee(iterable))
|
||
|
||
|
||
def polynomial_derivative(coefficients):
|
||
"""Compute the first derivative of a polynomial.
|
||
|
||
Evaluate the derivative of ``x³ - 4 x² - 17 x + 60``:
|
||
|
||
>>> coefficients = [1, -4, -17, 60]
|
||
>>> derivative_coefficients = polynomial_derivative(coefficients)
|
||
>>> derivative_coefficients
|
||
[3, -8, -17]
|
||
|
||
Note that polynomial coefficients are specified in descending power order.
|
||
|
||
Supports all numeric types: int, float, complex, Decimal, Fraction.
|
||
"""
|
||
n = len(coefficients)
|
||
powers = reversed(range(1, n))
|
||
return list(map(mul, coefficients, powers))
|
||
|
||
|
||
def totient(n):
|
||
"""Return the count of natural numbers up to *n* that are coprime with *n*.
|
||
|
||
Euler's totient function φ(n) gives the number of totatives.
|
||
Totative are integers k in the range 1 ≤ k ≤ n such that gcd(n, k) = 1.
|
||
|
||
>>> n = 9
|
||
>>> totient(n)
|
||
6
|
||
|
||
>>> totatives = [x for x in range(1, n) if gcd(n, x) == 1]
|
||
>>> totatives
|
||
[1, 2, 4, 5, 7, 8]
|
||
>>> len(totatives)
|
||
6
|
||
|
||
Reference: https://en.wikipedia.org/wiki/Euler%27s_totient_function
|
||
|
||
"""
|
||
for prime in set(factor(n)):
|
||
n -= n // prime
|
||
return n
|
||
|
||
|
||
# Miller–Rabin primality test: https://oeis.org/A014233
|
||
_perfect_tests = [
|
||
(2047, (2,)),
|
||
(9080191, (31, 73)),
|
||
(4759123141, (2, 7, 61)),
|
||
(1122004669633, (2, 13, 23, 1662803)),
|
||
(2152302898747, (2, 3, 5, 7, 11)),
|
||
(3474749660383, (2, 3, 5, 7, 11, 13)),
|
||
(18446744073709551616, (2, 325, 9375, 28178, 450775, 9780504, 1795265022)),
|
||
(
|
||
3317044064679887385961981,
|
||
(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41),
|
||
),
|
||
]
|
||
|
||
|
||
@lru_cache
|
||
def _shift_to_odd(n):
|
||
'Return s, d such that 2**s * d == n'
|
||
s = ((n - 1) ^ n).bit_length() - 1
|
||
d = n >> s
|
||
assert (1 << s) * d == n and d & 1 and s >= 0
|
||
return s, d
|
||
|
||
|
||
def _strong_probable_prime(n, base):
|
||
assert (n > 2) and (n & 1) and (2 <= base < n)
|
||
|
||
s, d = _shift_to_odd(n - 1)
|
||
|
||
x = pow(base, d, n)
|
||
if x == 1 or x == n - 1:
|
||
return True
|
||
|
||
for _ in range(s - 1):
|
||
x = x * x % n
|
||
if x == n - 1:
|
||
return True
|
||
|
||
return False
|
||
|
||
|
||
# Separate instance of Random() that doesn't share state
|
||
# with the default user instance of Random().
|
||
_private_randrange = random.Random().randrange
|
||
|
||
|
||
def is_prime(n):
|
||
"""Return ``True`` if *n* is prime and ``False`` otherwise.
|
||
|
||
Basic examples:
|
||
|
||
>>> is_prime(37)
|
||
True
|
||
>>> is_prime(3 * 13)
|
||
False
|
||
>>> is_prime(18_446_744_073_709_551_557)
|
||
True
|
||
|
||
Find the next prime over one billion:
|
||
|
||
>>> next(filter(is_prime, count(10**9)))
|
||
1000000007
|
||
|
||
Generate random primes up to 200 bits and up to 60 decimal digits:
|
||
|
||
>>> from random import seed, randrange, getrandbits
|
||
>>> seed(18675309)
|
||
|
||
>>> next(filter(is_prime, map(getrandbits, repeat(200))))
|
||
893303929355758292373272075469392561129886005037663238028407
|
||
|
||
>>> next(filter(is_prime, map(randrange, repeat(10**60))))
|
||
269638077304026462407872868003560484232362454342414618963649
|
||
|
||
This function is exact for values of *n* below 10**24. For larger inputs,
|
||
the probabilistic Miller-Rabin primality test has a less than 1 in 2**128
|
||
chance of a false positive.
|
||
"""
|
||
|
||
if n < 17:
|
||
return n in {2, 3, 5, 7, 11, 13}
|
||
|
||
if not (n & 1 and n % 3 and n % 5 and n % 7 and n % 11 and n % 13):
|
||
return False
|
||
|
||
for limit, bases in _perfect_tests:
|
||
if n < limit:
|
||
break
|
||
else:
|
||
bases = (_private_randrange(2, n - 1) for i in range(64))
|
||
|
||
return all(_strong_probable_prime(n, base) for base in bases)
|
||
|
||
|
||
def loops(n):
|
||
"""Returns an iterable with *n* elements for efficient looping.
|
||
Like ``range(n)`` but doesn't create integers.
|
||
|
||
>>> i = 0
|
||
>>> for _ in loops(5):
|
||
... i += 1
|
||
>>> i
|
||
5
|
||
|
||
"""
|
||
return repeat(None, n)
|
||
|
||
|
||
def multinomial(*counts):
|
||
"""Number of distinct arrangements of a multiset.
|
||
|
||
The expression ``multinomial(3, 4, 2)`` has several equivalent
|
||
interpretations:
|
||
|
||
* In the expansion of ``(a + b + c)⁹``, the coefficient of the
|
||
``a³b⁴c²`` term is 1260.
|
||
|
||
* There are 1260 distinct ways to arrange 9 balls consisting of 3 reds, 4
|
||
greens, and 2 blues.
|
||
|
||
* There are 1260 unique ways to place 9 distinct objects into three bins
|
||
with sizes 3, 4, and 2.
|
||
|
||
The :func:`multinomial` function computes the length of
|
||
:func:`distinct_permutations`. For example, there are 83,160 distinct
|
||
anagrams of the word "abracadabra":
|
||
|
||
>>> from more_itertools import distinct_permutations, ilen
|
||
>>> ilen(distinct_permutations('abracadabra'))
|
||
83160
|
||
|
||
This can be computed directly from the letter counts, 5a 2b 2r 1c 1d:
|
||
|
||
>>> from collections import Counter
|
||
>>> list(Counter('abracadabra').values())
|
||
[5, 2, 2, 1, 1]
|
||
>>> multinomial(5, 2, 2, 1, 1)
|
||
83160
|
||
|
||
A binomial coefficient is a special case of multinomial where there are
|
||
only two categories. For example, the number of ways to arrange 12 balls
|
||
with 5 reds and 7 blues is ``multinomial(5, 7)`` or ``math.comb(12, 5)``.
|
||
|
||
Likewise, factorial is a special case of multinomial where
|
||
the multiplicities are all just 1 so that
|
||
``multinomial(1, 1, 1, 1, 1, 1, 1) == math.factorial(7)``.
|
||
|
||
Reference: https://en.wikipedia.org/wiki/Multinomial_theorem
|
||
|
||
"""
|
||
return prod(map(comb, accumulate(counts), counts))
|
||
|
||
|
||
def _running_median_minheap_and_maxheap(iterator): # pragma: no cover
|
||
"Non-windowed running_median() for Python 3.14+"
|
||
|
||
read = iterator.__next__
|
||
lo = [] # max-heap
|
||
hi = [] # min-heap (same size as or one smaller than lo)
|
||
|
||
with suppress(StopIteration):
|
||
while True:
|
||
heappush_max(lo, heappushpop(hi, read()))
|
||
yield lo[0]
|
||
|
||
heappush(hi, heappushpop_max(lo, read()))
|
||
yield (lo[0] + hi[0]) / 2
|
||
|
||
|
||
def _running_median_minheap_only(iterator): # pragma: no cover
|
||
"Backport of non-windowed running_median() for Python 3.13 and prior."
|
||
|
||
read = iterator.__next__
|
||
lo = [] # max-heap (actually a minheap with negated values)
|
||
hi = [] # min-heap (same size as or one smaller than lo)
|
||
|
||
with suppress(StopIteration):
|
||
while True:
|
||
heappush(lo, -heappushpop(hi, read()))
|
||
yield -lo[0]
|
||
|
||
heappush(hi, -heappushpop(lo, -read()))
|
||
yield (hi[0] - lo[0]) / 2
|
||
|
||
|
||
def _running_median_windowed(iterator, maxlen):
|
||
"Yield median of values in a sliding window."
|
||
|
||
window = deque()
|
||
ordered = []
|
||
|
||
for x in iterator:
|
||
window.append(x)
|
||
insort(ordered, x)
|
||
|
||
if len(ordered) > maxlen:
|
||
i = bisect_left(ordered, window.popleft())
|
||
del ordered[i]
|
||
|
||
n = len(ordered)
|
||
m = n // 2
|
||
yield ordered[m] if n & 1 else (ordered[m - 1] + ordered[m]) / 2
|
||
|
||
|
||
def running_median(iterable, *, maxlen=None):
|
||
"""Cumulative median of values seen so far or values in a sliding window.
|
||
|
||
Set *maxlen* to a positive integer to specify the maximum size
|
||
of the sliding window. The default of *None* is equivalent to
|
||
an unbounded window.
|
||
|
||
For example:
|
||
|
||
>>> list(running_median([5.0, 9.0, 4.0, 12.0, 8.0, 9.0]))
|
||
[5.0, 7.0, 5.0, 7.0, 8.0, 8.5]
|
||
>>> list(running_median([5.0, 9.0, 4.0, 12.0, 8.0, 9.0], maxlen=3))
|
||
[5.0, 7.0, 5.0, 9.0, 8.0, 9.0]
|
||
|
||
Supports numeric types such as int, float, Decimal, and Fraction,
|
||
but not complex numbers which are unorderable.
|
||
|
||
On version Python 3.13 and prior, max-heaps are simulated with
|
||
negative values. The negation causes Decimal inputs to apply context
|
||
rounding, making the results slightly different than that obtained
|
||
by statistics.median().
|
||
"""
|
||
|
||
iterator = iter(iterable)
|
||
|
||
if maxlen is not None:
|
||
maxlen = index(maxlen)
|
||
if maxlen <= 0:
|
||
raise ValueError('Window size should be positive')
|
||
return _running_median_windowed(iterator, maxlen)
|
||
|
||
if not _max_heap_available:
|
||
return _running_median_minheap_only(iterator) # pragma: no cover
|
||
|
||
return _running_median_minheap_and_maxheap(iterator) # pragma: no cover
|
||
|
||
|
||
def _windowed_running_mean(iterator, n):
|
||
window = deque()
|
||
running_sum = 0
|
||
for value in iterator:
|
||
window.append(value)
|
||
running_sum += value
|
||
if len(window) > n:
|
||
running_sum -= window.popleft()
|
||
yield running_sum / len(window)
|
||
|
||
|
||
def running_mean(iterable, *, maxlen=None):
|
||
"""Cumulative mean of values seen so far or values in a sliding window.
|
||
|
||
Set *maxlen* to a positive integer to specify the maximum size
|
||
of the sliding window. The default of *None* is equivalent to
|
||
an unbounded window.
|
||
|
||
For example:
|
||
|
||
>>> list(running_mean([40, 30, 50, 46, 39, 44]))
|
||
[40.0, 35.0, 40.0, 41.5, 41.0, 41.5]
|
||
|
||
>>> list(running_mean([40, 30, 50, 46, 39, 44], maxlen=3))
|
||
[40.0, 35.0, 40.0, 42.0, 45.0, 43.0]
|
||
|
||
Supports numeric types such as int, float, complex, Decimal, and Fraction.
|
||
|
||
No extra effort is made to reduce round-off errors for float inputs.
|
||
So the results may be slightly different from `statistics.mean`.
|
||
|
||
"""
|
||
|
||
iterator = iter(iterable)
|
||
|
||
if maxlen is None:
|
||
return map(truediv, accumulate(iterator), count(1))
|
||
|
||
if maxlen <= 0:
|
||
raise ValueError('Window size should be positive')
|
||
|
||
return _windowed_running_mean(iterator, maxlen)
|
||
|
||
|
||
def random_derangement(iterable):
|
||
"""Return a random derangement of elements in the iterable.
|
||
|
||
Equivalent to but much faster than ``choice(list(derangements(iterable)))``.
|
||
|
||
"""
|
||
seq = tuple(iterable)
|
||
if len(seq) < 2:
|
||
if len(seq) == 0:
|
||
return ()
|
||
raise IndexError('No derangments to choose from')
|
||
perm = list(range(len(seq)))
|
||
start = tuple(perm)
|
||
while True:
|
||
shuffle(perm)
|
||
if not any(map(is_, start, perm)):
|
||
return itemgetter(*perm)(seq)
|