
from __future__ import annotations

from math import gcd






def 							_lowercase							(     UpperCamelCase_	,	UpperCamelCase_ = 2	,	UpperCamelCase_ = 1	,	UpperCamelCase_ = 3	,	)		->		int | None:
					'''simple docstring'''


					if num < 2:
										raise ValueError('The input value cannot be less than 2'		)

					# Because of the relationship between ``f(f(x))`` and ``f(x)``, this
					# algorithm struggles to find factors that are divisible by two.
					# As a workaround, we specifically check for two and even inputs.
					#   See: https://math.stackexchange.com/a/2856214/165820
					if num > 2 and num % 2 == 0:
										return 2

					# Pollard's Rho algorithm requires a function that returns pseudorandom
					# values between 0 <= X < ``num``.  It doesn't need to be random in the
					# sense that the output value is cryptographically secure or difficult
					# to calculate, it only needs to be random in the sense that all output
					# values should be equally likely to appear.
					# For this reason, Pollard suggested using ``f(x) = (x**2 - 1) % num``
					# However, the success of Pollard's algorithm isn't guaranteed and is
					# determined in part by the initial seed and the chosen random function.
					# To make retries easier, we will instead use ``f(x) = (x**2 + C) % num``
					# where ``C`` is a value that we can modify between each attempt.
					def rand_fn(UpperCamelCase_	,	UpperCamelCase_	,	UpperCamelCase_		) -> int:
										return (pow(UpperCamelCase_	,	2		) + step) % modulus

					for _ in range(UpperCamelCase_		):
										# These track the position within the cycle detection logic.
										SCREAMING_SNAKE_CASE__													= seed
										SCREAMING_SNAKE_CASE__													= seed

										while True:
															# At each iteration, the tortoise moves one step and the hare moves two.
															SCREAMING_SNAKE_CASE__													= rand_fn(UpperCamelCase_	,	UpperCamelCase_	,	UpperCamelCase_		)
															SCREAMING_SNAKE_CASE__													= rand_fn(UpperCamelCase_	,	UpperCamelCase_	,	UpperCamelCase_		)
															SCREAMING_SNAKE_CASE__													= rand_fn(UpperCamelCase_	,	UpperCamelCase_	,	UpperCamelCase_		)

															# At some point both the tortoise and the hare will enter a cycle whose
															# length ``p`` is a divisor of ``num``.  Once in that cycle, at some point
															# the tortoise and hare will end up on the same value modulo ``p``.
															# We can detect when this happens because the position difference between
															# the tortoise and the hare will share a common divisor with ``num``.
															SCREAMING_SNAKE_CASE__													= gcd(hare - tortoise	,	UpperCamelCase_		)

															if divisor == 1:
																				# No common divisor yet, just keep searching.
																				continue
															else:
																				# We found a common divisor!
																				if divisor == num:
																									# Unfortunately, the divisor is ``num`` itself and is useless.
																									break
																				else:
																									# The divisor is a nontrivial factor of ``num``!
																									return divisor

        # If we made it here, then this attempt failed.
        # We need to pick a new starting seed for the tortoise and hare
        # in addition to a new step value for the random function.
        # To keep this example implementation deterministic, the
        # new values will be generated based on currently available
        # values instead of using something like ``random.randint``.

        # We can use the hare's position as the new seed.
        # This is actually what Richard Brent's the "optimized" variant does.
										SCREAMING_SNAKE_CASE__													= hare

										# The new step value for the random function can just be incremented.
										# At first the results will be similar to what the old function would
										# have produced, but the value will quickly diverge after a bit.
										step += 1

					# We haven't found a divisor within the requested number of attempts.
					# We were unlucky or ``num`` itself is actually prime.
					return None


if __name__ == "__main__":
				import argparse

				__snake_case								=  argparse.ArgumentParser()
				parser.add_argument(
				    """num""",
				    type=int,
				    help="""The value to find a divisor of""",
				)
				parser.add_argument(
				    """--attempts""",
				    type=int,
				    default=3,
				    help="""The number of attempts before giving up""",
				)
				__snake_case								=  parser.parse_args()

				__snake_case								=  pollard_rho(args.num, attempts=args.attempts)
				if divisor is None:
								print(F"""{args.num} is probably prime""")
				else:
								__snake_case								=  args.num // divisor
								print(F"""{args.num} = {divisor} * {quotient}""")







from typing import List, Optional, TypeVar

from .arrow_dataset import Dataset, _concatenate_map_style_datasets, _interleave_map_style_datasets
from .dataset_dict import DatasetDict, IterableDatasetDict
from .info import DatasetInfo
from .iterable_dataset import IterableDataset, _concatenate_iterable_datasets, _interleave_iterable_datasets
from .splits import NamedSplit
from .utils import logging
from .utils.py_utils import Literal


__snake_case								=  logging.get_logger(__name__)


__snake_case								=  TypeVar("""DatasetType""", Dataset, IterableDataset)






def 							_lowercase							(     UpperCamelCase_	,	UpperCamelCase_ = None	,	UpperCamelCase_ = None	,	UpperCamelCase_ = None	,	UpperCamelCase_ = None	,	UpperCamelCase_ = "first_exhausted"	,	)		->		DatasetType:
					'''simple docstring'''


					from .arrow_dataset import Dataset
					from .iterable_dataset import IterableDataset

					if not datasets:
										raise ValueError('Unable to interleave an empty list of datasets.'		)
					for i, dataset in enumerate(UpperCamelCase_		):
										if not isinstance(UpperCamelCase_	,	(Dataset, IterableDataset)		):
															if isinstance(UpperCamelCase_	,	(DatasetDict, IterableDatasetDict)		):
																				if not dataset:
																									raise ValueError(
																									    F'Expected a list of Dataset objects or a list of IterableDataset objects, but element at position {i} '
																									    'is an empty dataset dictionary.'		)
																				raise ValueError(
																				    F'Dataset at position {i} has at least one split: {list(UpperCamelCase_		)}\n'
																				    F'Please pick one to interleave with the other datasets, for example: dataset[\'{next(iter(UpperCamelCase_		)		)}\']'		)
															raise ValueError(
															    F'Expected a list of Dataset objects or a list of IterableDataset objects, but element at position {i} is a {type(UpperCamelCase_		).__name__}.'		)
										if i == 0:
															SCREAMING_SNAKE_CASE__						, SCREAMING_SNAKE_CASE__													= (
															    (Dataset, IterableDataset) if isinstance(UpperCamelCase_	,	UpperCamelCase_		) else (IterableDataset, Dataset)
															)
										elif not isinstance(UpperCamelCase_	,	UpperCamelCase_		):
															raise ValueError(
															    F'Unable to interleave a {dataset_type.__name__} (at position 0) with a {other_type.__name__} (at position {i}). Expected a list of Dataset objects or a list of IterableDataset objects.'		)
					if stopping_strategy not in ["first_exhausted", "all_exhausted"]:
										raise ValueError(F'{stopping_strategy} is not supported. Please enter a valid stopping_strategy.'		)
					if dataset_type is Dataset:
										return _interleave_map_style_datasets(
										    UpperCamelCase_	,	UpperCamelCase_	,	UpperCamelCase_	,	info=UpperCamelCase_	,	split=UpperCamelCase_	,	stopping_strategy=UpperCamelCase_		)
					else:
										return _interleave_iterable_datasets(
										    UpperCamelCase_	,	UpperCamelCase_	,	UpperCamelCase_	,	info=UpperCamelCase_	,	split=UpperCamelCase_	,	stopping_strategy=UpperCamelCase_		)






def 							_lowercase							(     UpperCamelCase_	,	UpperCamelCase_ = None	,	UpperCamelCase_ = None	,	UpperCamelCase_ = 0	,	)		->		DatasetType:
					'''simple docstring'''


					if not dsets:
										raise ValueError('Unable to concatenate an empty list of datasets.'		)
					for i, dataset in enumerate(UpperCamelCase_		):
										if not isinstance(UpperCamelCase_	,	(Dataset, IterableDataset)		):
															if isinstance(UpperCamelCase_	,	(DatasetDict, IterableDatasetDict)		):
																				if not dataset:
																									raise ValueError(
																									    F'Expected a list of Dataset objects or a list of IterableDataset objects, but element at position {i} '
																									    'is an empty dataset dictionary.'		)
																				raise ValueError(
																				    F'Dataset at position {i} has at least one split: {list(UpperCamelCase_		)}\n'
																				    F'Please pick one to interleave with the other datasets, for example: dataset[\'{next(iter(UpperCamelCase_		)		)}\']'		)
															raise ValueError(
															    F'Expected a list of Dataset objects or a list of IterableDataset objects, but element at position {i} is a {type(UpperCamelCase_		).__name__}.'		)
										if i == 0:
															SCREAMING_SNAKE_CASE__						, SCREAMING_SNAKE_CASE__													= (
															    (Dataset, IterableDataset) if isinstance(UpperCamelCase_	,	UpperCamelCase_		) else (IterableDataset, Dataset)
															)
										elif not isinstance(UpperCamelCase_	,	UpperCamelCase_		):
															raise ValueError(
															    F'Unable to interleave a {dataset_type.__name__} (at position 0) with a {other_type.__name__} (at position {i}). Expected a list of Dataset objects or a list of IterableDataset objects.'		)
					if dataset_type is Dataset:
										return _concatenate_map_style_datasets(UpperCamelCase_	,	info=UpperCamelCase_	,	split=UpperCamelCase_	,	axis=UpperCamelCase_		)
					else:
										return _concatenate_iterable_datasets(UpperCamelCase_	,	info=UpperCamelCase_	,	split=UpperCamelCase_	,	axis=UpperCamelCase_		)





