
"""simple docstring"""





def       lowerCamelCase__		(			__snake_case	)							->		Any:


					"""simple docstring"""





					_UpperCamelCase							 =  0
					_UpperCamelCase							 =  len(__snake_case	)

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

					return num_inversions



def       lowerCamelCase__		(			__snake_case	)							->		List[str]:


					"""simple docstring"""





					if len(__snake_case	) <= 1:
										return arr, 0
					_UpperCamelCase							 =  len(__snake_case	) // 2
					_UpperCamelCase							 =  arr[0:mid]
					_UpperCamelCase							 =  arr[mid:]

					_UpperCamelCase ,  _UpperCamelCase							 =  count_inversions_recursive(__snake_case	)
					_UpperCamelCase ,  _UpperCamelCase							 =  count_inversions_recursive(__snake_case	)
					_UpperCamelCase ,  _UpperCamelCase							 =  _count_cross_inversions(__snake_case,     __snake_case	)

					_UpperCamelCase							 =  inversion_p + inversions_q + cross_inversions
					return c, num_inversions



def       lowerCamelCase__		(			__snake_case,     __snake_case	)							->		Any:


					"""simple docstring"""





					_UpperCamelCase							 =  []
					_UpperCamelCase							 =  _UpperCamelCase							 =  _UpperCamelCase							 =  0
					while i < len(__snake_case	) and j < len(__snake_case	):
										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(__snake_case	) - i
															r.append(q[j]	)
															j += 1
										else:
															r.append(p[i]	)
															i += 1

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

					return r, num_inversion



def       lowerCamelCase__		(			)							->		Optional[Any]:


					"""simple docstring"""





					_UpperCamelCase							 =  [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)

					_UpperCamelCase							 =  count_inversions_bf(__snake_case	)
					_UpperCamelCase ,  _UpperCamelCase							 =  count_inversions_recursive(__snake_case	)

					assert num_inversions_bf == num_inversions_recursive == 8

					print('''number of inversions = ''',     __snake_case	)

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

					arr_a.sort()
					_UpperCamelCase							 =  count_inversions_bf(__snake_case	)
					_UpperCamelCase ,  _UpperCamelCase							 =  count_inversions_recursive(__snake_case	)

					assert num_inversions_bf == num_inversions_recursive == 0
					print('''number of inversions = ''',     __snake_case	)

					# an empty list should also have zero inversions
					_UpperCamelCase							 =  []
					_UpperCamelCase							 =  count_inversions_bf(__snake_case	)
					_UpperCamelCase ,  _UpperCamelCase							 =  count_inversions_recursive(__snake_case	)

					assert num_inversions_bf == num_inversions_recursive == 0
					print('''number of inversions = ''',     __snake_case	)


if __name__ == "__main__":
				main()








"""simple docstring"""





from graphs.minimum_spanning_tree_kruskal import kruskal



def       lowerCamelCase__		(			)							->		List[Any]:


					"""simple docstring"""





					_UpperCamelCase							 =  9
					_UpperCamelCase							 =  [
					    [0, 1, 4],
					    [0, 7, 8],
					    [1, 2, 8],
					    [7, 8, 7],
					    [7, 6, 1],
					    [2, 8, 2],
					    [8, 6, 6],
					    [2, 3, 7],
					    [2, 5, 4],
					    [6, 5, 2],
					    [3, 5, 14],
					    [3, 4, 9],
					    [5, 4, 10],
					    [1, 7, 11],
					]

					_UpperCamelCase							 =  kruskal(__snake_case,     __snake_case	)

					_UpperCamelCase							 =  [
					    [7, 6, 1],
					    [2, 8, 2],
					    [6, 5, 2],
					    [0, 1, 4],
					    [2, 5, 4],
					    [2, 3, 7],
					    [0, 7, 8],
					    [3, 4, 9],
					]

					assert sorted(__snake_case	) == sorted(__snake_case	)






