






def     _a					(       UpperCamelCase_   :			list				)	->				bool:
						"""simple docstring"""







						if not isinstance(SCREAMING_SNAKE_CASE_      ,				SCREAMING_SNAKE_CASE_				):
												raise ValueError("Input series is not valid, valid series - [1, 2/3, 2]"				)
						if len(SCREAMING_SNAKE_CASE_				) == 0:
												raise ValueError("Input list must be a non empty list"				)
						if len(SCREAMING_SNAKE_CASE_				) == 1 and series[0] != 0:
												return True
						lowerCAmelCase__	 =  []
						lowerCAmelCase__	 =  len(SCREAMING_SNAKE_CASE_				)
						for i in range(0      ,				SCREAMING_SNAKE_CASE_				):
												if series[i] == 0:
																		raise ValueError("Input series cannot have 0 as an element"				)
												rec_series.append(1 / series[i]				)
						lowerCAmelCase__	 =  rec_series[1] - rec_series[0]
						for index in range(2      ,				SCREAMING_SNAKE_CASE_				):
												if rec_series[index] - rec_series[index - 1] != common_diff:
																		return False
						return True


def     _a					(       UpperCamelCase_   :			list				)	->				float:
						"""simple docstring"""







						if not isinstance(SCREAMING_SNAKE_CASE_      ,				SCREAMING_SNAKE_CASE_				):
												raise ValueError("Input series is not valid, valid series - [2, 4, 6]"				)
						if len(SCREAMING_SNAKE_CASE_				) == 0:
												raise ValueError("Input list must be a non empty list"				)
						lowerCAmelCase__	 =  0
						for val in series:
												answer += 1 / val
						return len(SCREAMING_SNAKE_CASE_				) / answer


if __name__ == "__main__":
				import doctest

				doctest.testmod()














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
a_							=      {
    '''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     _a					(       UpperCamelCase_   :			dict[str, list[str]]				)	->				set[tuple[str, str]]:
						"""simple docstring"""







						lowerCAmelCase__	 =  {node: {node} for node in graph}

						lowerCAmelCase__	 =  {node: graph[node][:] for node in graph}

						while len(UpperCamelCase_				) > 2:
												# Choose a random edge.
												lowerCAmelCase__	 =  random.choice(list(graph_copy.keys()				)				)
												lowerCAmelCase__	 =  random.choice(graph_copy[u]				)

												# Contract edge (u, v) to new node uv
												lowerCAmelCase__	 =  u + v
												lowerCAmelCase__	 =  list(set(graph_copy[u] + graph_copy[v]				)				)
												uv_neighbors.remove(UpperCamelCase_				)
												uv_neighbors.remove(UpperCamelCase_				)
												lowerCAmelCase__	 =  uv_neighbors
												for neighbor in uv_neighbors:
																		graph_copy[neighbor].append(UpperCamelCase_				)

												lowerCAmelCase__	 =  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(UpperCamelCase_				)
																		if v in graph_copy[neighbor]:
																								graph_copy[neighbor].remove(UpperCamelCase_				)

    # Find cutset.
						lowerCAmelCase__	 =  [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))






