

"""simple docstring"""






from __future__ import annotations

import random

# Adjacency list representation of this graph:
# https://en.wikipedia.org/wiki/File:Single_run_of_Karger%E2%80%99s_Mincut_algorithm.svg
lowerCamelCase__				:  Dict													=  {
    """1""": ["""2""", """3""", """4""", """5"""],
    """2""": ["""1""", """3""", """4""", """5"""],
    """3""": ["""1""", """2""", """4""", """5""", """10"""],
    """4""": ["""1""", """2""", """3""", """5""", """6"""],
    """5""": ["""1""", """2""", """3""", """4""", """7"""],
    """6""": ["""7""", """8""", """9""", """10""", """4"""],
    """7""": ["""6""", """8""", """9""", """10""", """5"""],
    """8""": ["""6""", """7""", """9""", """10"""],
    """9""": ["""6""", """7""", """8""", """10"""],
    """10""": ["""6""", """7""", """8""", """9""", """3"""],
}
def       UpperCamelCase					(			_lowerCAmelCase     :						Optional[Any] )				->			List[Any]:
	_UpperCAmelCase    :							List[Any]									=					{node: {node} for node in graph}

	_UpperCAmelCase    :							Optional[Any]									=					{node: graph[node][:] for node in graph}

	while len(_lowerCAmelCase ) > 2:
		# Choose a random edge.
		_UpperCAmelCase    :							Union[str, Any]									=					random.choice(list(graph_copy.keys() ) )
		_UpperCAmelCase    :							List[Any]									=					random.choice(graph_copy[u] )

		# Contract edge (u, v) to new node uv
		_UpperCAmelCase    :							str									=					u + v
		_UpperCAmelCase    :							Optional[int]									=					list(set(graph_copy[u] + graph_copy[v] ) )
		uv_neighbors.remove(_lowerCAmelCase )
		uv_neighbors.remove(_lowerCAmelCase )
		_UpperCAmelCase    :							Union[str, Any]									=					uv_neighbors
		for neighbor in uv_neighbors:
			graph_copy[neighbor].append(_lowerCAmelCase )

		_UpperCAmelCase    :							Optional[Any]									=					set(contracted_nodes[u].union(contracted_nodes[v] ) )

		# Remove nodes u and v.
		del graph_copy[u]
		del graph_copy[v]
		for neighbor in uv_neighbors:
			if u in graph_copy[neighbor]:
				graph_copy[neighbor].remove(_lowerCAmelCase )
			if v in graph_copy[neighbor]:
				graph_copy[neighbor].remove(_lowerCAmelCase )

    # Find cutset.
	_UpperCAmelCase    :							List[Any]									=					[contracted_nodes[node] for node in graph_copy]
	return {
	    (node, neighbor)
	    for node in groups[0]
	    for neighbor in graph[node]
	    if neighbor in groups[1]
	}


if __name__ == "__main__":
						print(partition_graph(TEST_GRAPH))



"""simple docstring"""






from __future__ import annotations
def       UpperCamelCase					(			_lowerCAmelCase     :						list,  _lowerCAmelCase     :						list )				->			list:
	_UpperCAmelCase    :							List[Any]									=					[None] * (len(_lowerCAmelCase ) + len(_lowerCAmelCase ))

	_UpperCAmelCase    :							Union[str, Any]									=					0  # pointer to current index for left Half
	_UpperCAmelCase    :							str									=					0  # pointer to current index for the right Half
	_UpperCAmelCase    :							Optional[int]									=					0  # pointer to current index for the sorted array Half

	while pointera < len(_lowerCAmelCase ) and pointera < len(_lowerCAmelCase ):
		if left_half[pointera] < right_half[pointera]:
			_UpperCAmelCase    :							Tuple									=					left_half[pointera]
			pointera += 1
			index += 1
		else:
			_UpperCAmelCase    :							int									=					right_half[pointera]
			pointera += 1
			index += 1
	while pointera < len(_lowerCAmelCase ):
		_UpperCAmelCase    :							Tuple									=					left_half[pointera]
		pointera += 1
		index += 1

	while pointera < len(_lowerCAmelCase ):
		_UpperCAmelCase    :							Union[str, Any]									=					right_half[pointera]
		pointera += 1
		index += 1

	return sorted_array
def       UpperCamelCase					(			_lowerCAmelCase     :						list )				->			list:
	if len(_lowerCAmelCase ) <= 1:
		return array
	# the actual formula to calculate the middle element = left + (right - left) // 2
	# this avoids integer overflow in case of large N
	_UpperCAmelCase    :							Any									=					0 + (len(_lowerCAmelCase ) - 0) // 2

	# Split the array into halves till the array length becomes equal to One
	# merge the arrays of single length returned by mergeSort function and
	# pass them into the merge arrays function which merges the array
	_UpperCAmelCase    :							Optional[Any]									=					array[:middle]
	_UpperCAmelCase    :							Optional[int]									=					array[middle:]

	return merge(merge_sort(_lowerCAmelCase ),  merge_sort(_lowerCAmelCase ) )


if __name__ == "__main__":
						import doctest

						doctest.testmod()





