





def  __magic_name__  (				lowercase				)				-> str:

							"""simple docstring"""



							lowercase_	:   Optional[int]  				=     0
							lowercase_	:   Optional[int]  				=     len(lowercase				)

							for i in range(n - 1				):
														for j in range(i + 1				,       lowercase				):
																					if arr[i] > arr[j]:
																												num_inversions += 1

							return num_inversions





def  __magic_name__  (				lowercase				)				-> int:

							"""simple docstring"""



							if len(lowercase				) <= 1:
														return arr, 0
							lowercase_	:   Optional[Any]  				=     len(lowercase				) // 2
							lowercase_	:   Optional[int]  				=     arr[0:mid]
							lowercase_	:   Any  				=     arr[mid:]

							lowercase_				,		lowercase_	:   List[Any]  				=     count_inversions_recursive(lowercase				)
							lowercase_				,		lowercase_	:   Tuple  				=     count_inversions_recursive(lowercase				)
							lowercase_				,		lowercase_	:   str  				=     _count_cross_inversions(lowercase				,       lowercase				)

							lowercase_	:   Tuple  				=     inversion_p + inversions_q + cross_inversions
							return c, num_inversions





def  __magic_name__  (				lowercase				,       lowercase				)				-> int:

							"""simple docstring"""



							lowercase_	:   str  				=     []
							lowercase_	:   str  				=     0
							while i < len(lowercase				) and j < len(lowercase				):
														if p[i] > q[j]:
																					# if P[1] > Q[j], then P[k] > Q[k] for all  i < k <= len(P)
																					# These are all inversions. The claim emerges from the
																					# property that P is sorted.
																					num_inversion += len(lowercase				) - i
																					r.append(q[j]				)
																					j += 1
														else:
																					r.append(p[i]				)
																					i += 1

							if i < len(lowercase				):
														r.extend(p[i:]				)
							else:
														r.extend(q[j:]				)

							return r, num_inversion





def  __magic_name__  (				)				-> Optional[int]:

							"""simple docstring"""



							lowercase_	:   Union[str, Any]  				=     [10, 2, 1, 5, 5, 2, 11]

							# this arr has 8 inversions:
							# (10, 2), (10, 1), (10, 5), (10, 5), (10, 2), (2, 1), (5, 2), (5, 2)

							lowercase_	:   Dict  				=     count_inversions_bf(lowercase				)
							lowercase_				,		lowercase_	:   Optional[Any]  				=     count_inversions_recursive(lowercase				)

							assert num_inversions_bf == num_inversions_recursive == 8

							print("""number of inversions = """				,       lowercase				)

							# testing an array with zero inversion (a sorted arr_1)

							arr_a.sort()
							lowercase_	:   List[Any]  				=     count_inversions_bf(lowercase				)
							lowercase_				,		lowercase_	:   Any  				=     count_inversions_recursive(lowercase				)

							assert num_inversions_bf == num_inversions_recursive == 0
							print("""number of inversions = """				,       lowercase				)

							# an empty list should also have zero inversions
							lowercase_	:   str  				=     []
							lowercase_	:   str  				=     count_inversions_bf(lowercase				)
							lowercase_				,		lowercase_	:   int  				=     count_inversions_recursive(lowercase				)

							assert num_inversions_bf == num_inversions_recursive == 0
							print("""number of inversions = """				,       lowercase				)


if __name__ == "__main__":
							main()






UpperCAmelCase_							=				{"""a""": ["""c""", """b"""], """b""": ["""d""", """e"""], """c""": [], """d""": [], """e""": []}
UpperCAmelCase_							=				["""a""", """b""", """c""", """d""", """e"""]





def  __magic_name__  (				lowercase				,       lowercase				,       lowercase				)				-> Union[str, Any]:

							"""simple docstring"""



							lowercase_	:   Tuple  				=     start
							# add current to visited
							visited.append(lowercase				)
							lowercase_	:   Any  				=     edges[current]
							for neighbor in neighbors:
														# if neighbor not in visited, visit
														if neighbor not in visited:
																					lowercase_	:   List[str]  				=     topological_sort(lowercase				,       lowercase				,       lowercase				)
    # if all neighbors visited add current to sort
							sort.append(lowercase				)
							# if all vertices haven't been visited select a new one to visit
							if len(lowercase				) != len(lowercase				):
														for vertice in vertices:
																					if vertice not in visited:
																												lowercase_	:   Optional[Any]  				=     topological_sort(lowercase				,       lowercase				,       lowercase				)
    # return sort
							return sort


if __name__ == "__main__":
							UpperCAmelCase_							=				topological_sort("""a""", [], [])
							print(sort)