




def __UpperCamelCase			(       _A ,  _A ,  _A ,  _A  ):




						if graph[path[curr_ind - 1]][next_ver] == 0:
												return False

						# 2. Validate that next vertex is not already in path
						return not any(vertex == next_ver for vertex in path  )





def __UpperCamelCase			(       _A ,  _A ,  _A  ):




						if curr_ind == len(lowerCAmelCase__  ):
												# return whether path exists between current and starting vertices
												return graph[path[curr_ind - 1]][path[0]] == 1

						# Recursive Step
						for next_ver in range(0 ,  len(lowerCAmelCase__  )  ):
												if valid_connection(lowerCAmelCase__ ,  lowerCAmelCase__ ,  lowerCAmelCase__ ,  lowerCAmelCase__  ):
																		# Insert current vertex  into path as next transition
																		lowerCAmelCase_      			=  next_ver
																		# Validate created path
																		if util_hamilton_cycle(lowerCAmelCase__ ,  lowerCAmelCase__ ,  curr_ind + 1  ):
																								return True
																		# Backtrack
																		lowerCAmelCase_      			=  -1
						return False






def __UpperCamelCase			(       _A ,  _A = 0  ):
						lowerCAmelCase_      			=  [-1] * (len(lowerCAmelCase__  ) + 1)
						# initialize start and end of path with starting index
						lowerCAmelCase_      			=  start_index
						# evaluate and if we find answer return path either return empty array
						return path if util_hamilton_cycle(lowerCAmelCase__ ,  lowerCAmelCase__ ,  1  ) else []









"""simple docstring"""







import random

def 				UpperCamelCase_    (    lowerCAmelCase__      :				int    ,   lowerCAmelCase__      :				float    ,   lowerCAmelCase__      :				bool = False						)						->   dict:

  """simple docstring"""




  lowerCAmelCase_   :				dict								= {i: [] for i in range(lowerCAmelCase__						)}

  # if probability is greater or equal than 1, then generate a complete graph
  if probability >= 1:
    return complete_graph(lowerCAmelCase__						)
  # if probability is lower or equal than 0, then return a graph without edges
  if probability <= 0:
    return graph

  # for each couple of nodes, add an edge from u to v
  # if the number randomly generated is greater than probability probability
  for i in range(lowerCAmelCase__						):
    for j in range(i + 1    ,   lowerCAmelCase__						):
      if random.random() < probability:
        graph[i].append(lowerCAmelCase__						)
        if not directed:
          # if the graph is undirected, add an edge in from j to i, either
          graph[j].append(lowerCAmelCase__						)
  return graph

def 				UpperCamelCase_    (    lowerCAmelCase__      :				int						)						->   dict:

  """simple docstring"""




  return {
      i: [j for j in range(lowerCAmelCase__						) if i != j] for i in range(lowerCAmelCase__						)
  }


if __name__ == "__main__":
     import doctest

     doctest.testmod()





