



def __lowerCAmelCase			(  SCREAMING_SNAKE_CASE_	):
			lowercase__       	=					0
			lowercase__       	=					len(SCREAMING_SNAKE_CASE_	)

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

			return num_inversions



def __lowerCAmelCase			(  SCREAMING_SNAKE_CASE_	):
			if len(SCREAMING_SNAKE_CASE_	) <= 1:
						return arr, 0
			lowercase__       	=					len(SCREAMING_SNAKE_CASE_	) // 2
			lowercase__       	=					arr[0:mid]
			lowercase__       	=					arr[mid:]

			lowercase__	,							lowercase__       	=					count_inversions_recursive(SCREAMING_SNAKE_CASE_	)
			lowercase__	,							lowercase__       	=					count_inversions_recursive(SCREAMING_SNAKE_CASE_	)
			lowercase__	,							lowercase__       	=					_count_cross_inversions(SCREAMING_SNAKE_CASE_    ,	SCREAMING_SNAKE_CASE_	)

			lowercase__       	=					inversion_p + inversions_q + cross_inversions
			return c, num_inversions



def __lowerCAmelCase			(  SCREAMING_SNAKE_CASE_    ,	SCREAMING_SNAKE_CASE_	):
			lowercase__       	=					[]
			lowercase__       	=					lowercase__       	=					lowercase__       	=					0
			while i < len(SCREAMING_SNAKE_CASE_	) and j < len(SCREAMING_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(SCREAMING_SNAKE_CASE_	) - i
									r.append(q[j]	)
									j += 1
						else:
									r.append(p[i]	)
									i += 1

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

			return r, num_inversion



def __lowerCAmelCase			(  ):
			lowercase__       	=					[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__       	=					count_inversions_bf(SCREAMING_SNAKE_CASE_	)
			lowercase__	,							lowercase__       	=					count_inversions_recursive(SCREAMING_SNAKE_CASE_	)

			assert num_inversions_bf == num_inversions_recursive == 8

			print("number of inversions = "    ,	SCREAMING_SNAKE_CASE_	)

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

			arr_a.sort()
			lowercase__       	=					count_inversions_bf(SCREAMING_SNAKE_CASE_	)
			lowercase__	,							lowercase__       	=					count_inversions_recursive(SCREAMING_SNAKE_CASE_	)

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

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

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


if __name__ == "__main__":
							main()







'''simple docstring'''




def    lowerCamelCase__		(    SCREAMING_SNAKE_CASE   : list[int]							, SCREAMING_SNAKE_CASE   : list[int]							, SCREAMING_SNAKE_CASE   : int    ):
  return not any(
      neighbour == 1 and colored_vertices[i] == color
      for i, neighbour in enumerate(SCREAMING_SNAKE_CASE    )    )


def    lowerCamelCase__		(    SCREAMING_SNAKE_CASE   : list[list[int]]							, SCREAMING_SNAKE_CASE   : int							, SCREAMING_SNAKE_CASE   : list[int]							, SCREAMING_SNAKE_CASE   : int    ):

  # Base Case
  if index == len(SCREAMING_SNAKE_CASE    ):
    return True

  # Recursive Step
  for i in range(SCREAMING_SNAKE_CASE    ):
    if valid_coloring(graph[index]							, SCREAMING_SNAKE_CASE							, SCREAMING_SNAKE_CASE    ):
      # Color current vertex
      UpperCAmelCase    				=						i
      # Validate coloring
      if util_color(SCREAMING_SNAKE_CASE							, SCREAMING_SNAKE_CASE							, SCREAMING_SNAKE_CASE							, index + 1    ):
        return True
      # Backtrack
      UpperCAmelCase    				=						-1
  return False



def    lowerCamelCase__		(    SCREAMING_SNAKE_CASE   : list[list[int]]							, SCREAMING_SNAKE_CASE   : int    ):
  UpperCAmelCase    				=						[-1] * len(SCREAMING_SNAKE_CASE    )

  if util_color(SCREAMING_SNAKE_CASE							, SCREAMING_SNAKE_CASE							, SCREAMING_SNAKE_CASE							, 0    ):
    return colored_vertices

  return []


