


def        _UpperCamelCase							(      lowercase__      ):
						__SCREAMING_SNAKE_CASE						:     Union[str, Any]					=		set()
						# edges = list of graph's edges
						__SCREAMING_SNAKE_CASE						:     List[str]					=		get_edges(lowercase__      )

						# While there are still elements in edges list, take an arbitrary edge
						# (from_node, to_node) and add his extremity to chosen_vertices and then
						# remove all arcs adjacent to the from_node and to_node
						while edges:
												__SCREAMING_SNAKE_CASE			,			__SCREAMING_SNAKE_CASE						:     int					=		edges.pop()
												chosen_vertices.add(lowercase__      )
												chosen_vertices.add(lowercase__      )
												for edge in edges.copy():
																		if from_node in edge or to_node in edge:
																								edges.discard(lowercase__      )
						return chosen_vertices


def        _UpperCamelCase							(      lowercase__      ):
						__SCREAMING_SNAKE_CASE						:     Optional[int]					=		set()
						for from_node, to_nodes in graph.items():
												for to_node in to_nodes:
																		edges.add((from_node, to_node)      )
						return edges


if __name__ == "__main__":
						import doctest

						doctest.testmod()

						# graph = {0: [1, 3], 1: [0, 3], 2: [0, 3, 4], 3: [0, 1, 2], 4: [2, 3]}
						# print(f"Matching vertex cover:\n{matching_min_vertex_cover(graph)}")




import math_equivalence  # From: git+https://github.com/hendrycks/math.git

import datasets


__lowerCAmelCase	: Optional[Any]  				='\\n@article{hendrycksmath2021,\n  title={Measuring Mathematical Problem Solving With the MATH Dataset},\n  author={Dan Hendrycks\n    and Collin Burns\n    and Saurav Kadavath\n    and Akul Arora\n    and Steven Basart\n    and Eric Tang\n    and Dawn Song\n    and Jacob Steinhardt},\n  journal={arXiv preprint arXiv:2103.03874},\n  year={2021}\n}\n'


__lowerCAmelCase	: Any  				='\\nThis metric is used to assess performance on the Mathematics Aptitude Test of Heuristics (MATH) dataset.\nIt first canonicalizes the inputs (e.g., converting "1/2" to "\\frac{1}{2}") and then computes accuracy.\n'


__lowerCAmelCase	: Optional[Any]  				=r'\nCalculates accuracy after canonicalizing inputs.\n\nArgs:\n    predictions: list of predictions to score. Each prediction\n        is a string that contains natural language and LaTex.\n    references: list of reference for each prediction. Each\n        reference is a string that contains natural language\n        and LaTex.\nReturns:\n    accuracy: accuracy after canonicalizing inputs\n        (e.g., converting "1/2" to "\\frac{1}{2}")\n\nExamples:\n    >>> metric = datasets.load_metric("competition_math")\n    >>> results = metric.compute(references=["\\frac{1}{2}"], predictions=["1/2"])\n    >>> print(results)\n    {\'accuracy\': 1.0}\n'
@datasets.utils.file_utils.add_end_docstrings(_DESCRIPTION						,			_KWARGS_DESCRIPTION       )
class        _lowercase      (			datasets.Metric       ):






				'''simple docstring'''



				def 		__magic_name__(       self			:Optional[Any]	)		->			List[str]:
										return datasets.MetricInfo(
										    description=_DESCRIPTION			,			citation=_CITATION			,			inputs_description=_KWARGS_DESCRIPTION			,			features=datasets.Features(
										        {
										            '''predictions''': datasets.Value('''string'''	),
										            '''references''': datasets.Value('''string'''	),
										        }	)			,			homepage='''https://github.com/hendrycks/math'''			,			codebase_urls=['''https://github.com/hendrycks/math''']			,			)



				def 		__magic_name__(       self			:Any			,			lowerCAmelCase__			:Optional[int]			,			lowerCAmelCase__			:Any	)		->			Optional[int]:
										__SCREAMING_SNAKE_CASE						:     Tuple					=		0.0
										for i, j in zip(lowerCAmelCase__			,			lowerCAmelCase__	):
																n_correct += 1.0 if math_equivalence.is_equiv(lowerCAmelCase__			,			lowerCAmelCase__	) else 0.0
										__SCREAMING_SNAKE_CASE						:     str					=		n_correct / len(lowerCAmelCase__	)
										return {
										    "accuracy": accuracy,
										}
