



from typing import List

import numpy as np






def        A       (      _lowerCamelCase						):





							'''simple docstring'''




							_lowerCAmelCase  :							List[Any]				     =					{key: len(_lowerCamelCase						) for key, value in gen_kwargs.items() if isinstance(_lowerCamelCase       ,				_lowerCamelCase						)}
							if len(set(lists_lengths.values()						)						) > 1:
														raise RuntimeError(
														    (
														        "Sharding is ambiguous for this dataset: "
														        + "we found several data sources lists of different lengths, and we don't know over which list we should parallelize:\n"
														        + "\n".join(F"\t- key {key} has length {length}" for key, length in lists_lengths.items()						)
														        + "\nTo fix this, check the 'gen_kwargs' and make sure to use lists only for data sources, "
														        + "and use tuples otherwise. In the end there should only be one single list, or several lists with the same length."
														    )						)
							_lowerCAmelCase  :							Dict				     =					max(lists_lengths.values()       ,				default=0						)
							return max(1       ,				_lowerCamelCase						)






def        A       (      _lowerCamelCase       ,				_lowerCamelCase						):





							'''simple docstring'''




							_lowerCAmelCase  :							List[str]				     =					[]
							for group_idx in range(_lowerCamelCase						):
														_lowerCAmelCase  :							Optional[Any]				     =					num_shards // max_num_jobs + (group_idx < (num_shards % max_num_jobs))
														if num_shards_to_add == 0:
																					break
														_lowerCAmelCase  :							Optional[int]				     =					shards_indices_per_group[-1].stop if shards_indices_per_group else 0
														_lowerCAmelCase  :							Optional[int]				     =					range(_lowerCamelCase       ,				start + num_shards_to_add						)
														shards_indices_per_group.append(_lowerCamelCase						)
							return shards_indices_per_group






def        A       (      _lowerCamelCase       ,				_lowerCamelCase						):





							'''simple docstring'''




							_lowerCAmelCase  :							Optional[Any]				     =					_number_of_shards_in_gen_kwargs(_lowerCamelCase						)
							if num_shards == 1:
														return [dict(_lowerCamelCase						)]
							else:
														_lowerCAmelCase  :							Tuple				     =					_distribute_shards(num_shards=_lowerCamelCase       ,				max_num_jobs=_lowerCamelCase						)
														return [
														    {
														        key: [value[shard_idx] for shard_idx in shard_indices_per_group[group_idx]]
														        if isinstance(_lowerCamelCase       ,				_lowerCamelCase						)
														        else value
														        for key, value in gen_kwargs.items()
														    }
														    for group_idx in range(len(_lowerCamelCase						)						)
														]






def        A       (      _lowerCamelCase						):





							'''simple docstring'''




							return {
							    key: [value for gen_kwargs in gen_kwargs_list for value in gen_kwargs[key]]
							    if isinstance(gen_kwargs_list[0][key]       ,				_lowerCamelCase						)
							    else gen_kwargs_list[0][key]
							    for key in gen_kwargs_list[0]
							}






def        A       (      _lowerCamelCase       ,				_lowerCamelCase						):





							'''simple docstring'''




							_lowerCAmelCase  :							int				     =					{len(_lowerCamelCase						) for value in gen_kwargs.values() if isinstance(_lowerCamelCase       ,				_lowerCamelCase						)}
							_lowerCAmelCase  :							List[Any]				     =					{}
							for size in list_sizes:
														_lowerCAmelCase  :							int				     =					list(range(_lowerCamelCase						)						)
														rng.shuffle(indices_per_size[size]						)
							# Now let's copy the gen_kwargs and shuffle the lists based on their sizes
							_lowerCAmelCase  :							List[str]				     =					dict(_lowerCamelCase						)
							for key, value in shuffled_kwargs.items():
														if isinstance(_lowerCamelCase       ,				_lowerCamelCase						):
																					_lowerCAmelCase  :							Dict				     =					[value[i] for i in indices_per_size[len(_lowerCamelCase						)]]
							return shuffled_kwargs





from __future__ import annotations

from collections.abc import MutableSequence



class 						UpperCAmelCase_					:







							def __init__(     self,    __a,    __a):

														'''simple docstring'''

														if len(__a) != degree + 1:
																					raise ValueError(
																					    "The number of coefficients should be equal to the degree + 1.")

														_lowerCAmelCase  :							list[float]				     =					list(__a)
														_lowerCAmelCase  :							Any				     =					degree







							def __add__(     self,    __a):

														'''simple docstring'''

														if self.degree > polynomial_a.degree:
																					_lowerCAmelCase  :							Dict				     =					self.coefficients[:]
																					for i in range(polynomial_a.degree + 1):
																												coefficients[i] += polynomial_a.coefficients[i]
																					return Polynomial(self.degree,    __a)
														else:
																					_lowerCAmelCase  :							Union[str, Any]				     =					polynomial_a.coefficients[:]
																					for i in range(self.degree + 1):
																												coefficients[i] += self.coefficients[i]
																					return Polynomial(polynomial_a.degree,    __a)







							def __sub__(     self,    __a):

														'''simple docstring'''

														return self + polynomial_a * Polynomial(0,    [-1])







							def __neg__(     self):

														'''simple docstring'''

														return Polynomial(self.degree,    [-c for c in self.coefficients])







							def __mul__(     self,    __a):

														'''simple docstring'''

														_lowerCAmelCase  :							list[float]				     =					[0] * (self.degree + polynomial_a.degree + 1)
														for i in range(self.degree + 1):
																					for j in range(polynomial_a.degree + 1):
																												coefficients[i + j] += (
																												    self.coefficients[i] * polynomial_a.coefficients[j]
																												)

														return Polynomial(self.degree + polynomial_a.degree,    __a)







							def      snake_case__   (     self,    __a):

														'''simple docstring'''

														_lowerCAmelCase  :							int | float				     =					0
														for i in range(self.degree + 1):
																					result += self.coefficients[i] * (substitution**i)
														return result







							def __str__(     self):

														'''simple docstring'''

														_lowerCAmelCase  :							List[str]				     =					""
														for i in range(self.degree,    -1,    -1):
																					if self.coefficients[i] == 0:
																												continue
																					elif self.coefficients[i] > 0:
																												if polynomial:
																																			polynomial += " + "
																					else:
																												polynomial += " - "

																					if i == 0:
																												polynomial += str(abs(self.coefficients[i]))
																					elif i == 1:
																												polynomial += str(abs(self.coefficients[i])) + "x"
																					else:
																												polynomial += str(abs(self.coefficients[i])) + "x^" + str(__a)

														return polynomial







							def __repr__(     self):

														'''simple docstring'''

														return self.__str__()







							def      snake_case__   (     self):

														'''simple docstring'''

														_lowerCAmelCase  :							list[float]				     =					[0] * self.degree
														for i in range(self.degree):
																					_lowerCAmelCase  :							List[Any]				     =					self.coefficients[i + 1] * (i + 1)
														return Polynomial(self.degree - 1,    __a)







							def      snake_case__   (     self,    __a = 0):

														'''simple docstring'''

														_lowerCAmelCase  :							list[float]				     =					[0] * (self.degree + 2)
														_lowerCAmelCase  :							Optional[Any]				     =					constant
														for i in range(self.degree + 1):
																					_lowerCAmelCase  :							Dict				     =					self.coefficients[i] / (i + 1)
														return Polynomial(self.degree + 1,    __a)







							def __eq__(     self,    __a):

														'''simple docstring'''

														if not isinstance(__a,    __a):
																					return False

														if self.degree != polynomial_a.degree:
																					return False

														for i in range(self.degree + 1):
																					if self.coefficients[i] != polynomial_a.coefficients[i]:
																												return False

														return True







							def __ne__(     self,    __a):

														'''simple docstring'''

														return not self.__eq__(__a)
