






"""simple docstring"""







# Algorithm for the pigeonhole sorting








def       _lowerCAmelCase       (      lowerCAmelCase		):







       '''simple docstring'''
       UpperCAmelCase	    =			min(lowerCAmelCase		)  # min() finds the minimum value
       UpperCAmelCase	    =			max(lowerCAmelCase		)  # max() finds the maximum value

       UpperCAmelCase	    =			max_val - min_val + 1  # size is difference of max and min values plus one

       # list of pigeonholes of size equal to the variable size
       UpperCAmelCase	    =			[0] * size

       # Populate the pigeonholes.
       for x in a:
              assert isinstance(lowerCAmelCase	,      lowerCAmelCase		), "integers only please"
              holes[x - min_val] += 1

       # Putting the elements back into the array in an order.
       UpperCAmelCase	    =			0
       for count in range(lowerCAmelCase		):
              while holes[count] > 0:
                     holes[count] -= 1
                     UpperCAmelCase	    =			count + min_val
                     i += 1






def       _lowerCAmelCase       (      ):







       '''simple docstring'''
       UpperCAmelCase	    =			[8, 3, 2, 7, 4, 6, 8]
       pigeonhole_sort(lowerCAmelCase		)
       print("""Sorted order is:"""	,      """ """.join(lowerCAmelCase		)		)


if __name__ == "__main__":
 main()








"""simple docstring"""







def       _lowerCAmelCase       (      lowerCAmelCase		):







       '''simple docstring'''
       UpperCAmelCase	    =			0
       UpperCAmelCase	    =			len(lowerCAmelCase		)

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

       return num_inversions






def       _lowerCAmelCase       (      lowerCAmelCase		):







       '''simple docstring'''
       if len(lowerCAmelCase		) <= 1:
              return arr, 0
       UpperCAmelCase	    =			len(lowerCAmelCase		) // 2
       UpperCAmelCase	    =			arr[0:mid]
       UpperCAmelCase	    =			arr[mid:]

       UpperCAmelCase    ,       UpperCAmelCase	    =			count_inversions_recursive(lowerCAmelCase		)
       UpperCAmelCase    ,       UpperCAmelCase	    =			count_inversions_recursive(lowerCAmelCase		)
       UpperCAmelCase    ,       UpperCAmelCase	    =			_count_cross_inversions(lowerCAmelCase	,      lowerCAmelCase		)

       UpperCAmelCase	    =			inversion_p + inversions_q + cross_inversions
       return c, num_inversions






def       _lowerCAmelCase       (      lowerCAmelCase	,      lowerCAmelCase		):







       '''simple docstring'''
       UpperCAmelCase	    =			[]
       UpperCAmelCase	    =			UpperCAmelCase	    =			UpperCAmelCase	    =			0
       while i < len(lowerCAmelCase		) and j < len(lowerCAmelCase		):
              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(lowerCAmelCase		) - i
                     r.append(q[j]		)
                     j += 1
              else:
                     r.append(p[i]		)
                     i += 1

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

       return r, num_inversion






def       _lowerCAmelCase       (      ):







       '''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(lowerCAmelCase		)
       UpperCAmelCase    ,       UpperCAmelCase	    =			count_inversions_recursive(lowerCAmelCase		)

       assert num_inversions_bf == num_inversions_recursive == 8

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

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

       arr_a.sort()
       UpperCAmelCase	    =			count_inversions_bf(lowerCAmelCase		)
       UpperCAmelCase    ,       UpperCAmelCase	    =			count_inversions_recursive(lowerCAmelCase		)

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

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

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


if __name__ == "__main__":
 main()
