


class   __a			:
 def __init__(   self				, lowerCAmelCase__				, lowerCAmelCase__ )					->				str:




        '''simple docstring'''
        lowercase__:   Dict  			=   name
        lowercase__:   Dict  			=   val
 def __str__(   self )					->				Dict:




        '''simple docstring'''
        return F'{self.__class__.__name__}({self.name}, {self.val})'
 def __lt__(   self				, lowerCAmelCase__ )					->				int:




        '''simple docstring'''
        return self.val < other.val





class   __a			:
 def __init__(   self				, lowerCAmelCase__ )					->				Tuple:




        '''simple docstring'''
        lowercase__:   Optional[int]  			=   {}
        lowercase__:   Any  			=   {}
        lowercase__:   Any  			=   self.build_heap(lowerCAmelCase__ )
 def __getitem__(   self				, lowerCAmelCase__ )					->				Union[str, Any]:




        '''simple docstring'''
        return self.get_value(lowerCAmelCase__ )
 def 		SCREAMING_SNAKE_CASE__   (   self				, lowerCAmelCase__ )					->				Union[str, Any]:




        '''simple docstring'''
        return (idx - 1) // 2
 def 		SCREAMING_SNAKE_CASE__   (   self				, lowerCAmelCase__ )					->				Dict:




        '''simple docstring'''
        return idx * 2 + 1
 def 		SCREAMING_SNAKE_CASE__   (   self				, lowerCAmelCase__ )					->				Tuple:




        '''simple docstring'''
        return idx * 2 + 2
 def 		SCREAMING_SNAKE_CASE__   (   self				, lowerCAmelCase__ )					->				List[str]:




        '''simple docstring'''
        return self.heap_dict[key]
 def 		SCREAMING_SNAKE_CASE__   (   self				, lowerCAmelCase__ )					->				List[str]:




        '''simple docstring'''
        lowercase__:   List[Any]  			=   len(lowerCAmelCase__ ) - 1
        lowercase__:   int  			=   self.get_parent_idx(lowerCAmelCase__ )

        for idx, i in enumerate(lowerCAmelCase__ ):
               lowercase__:   Any  			=   idx
               lowercase__:   List[Any]  			=   i.val

        for i in range(lowerCAmelCase__				, -1				, -1 ):
               self.sift_down(lowerCAmelCase__				, lowerCAmelCase__ )
        return array
 def 		SCREAMING_SNAKE_CASE__   (   self				, lowerCAmelCase__				, lowerCAmelCase__ )					->				str:




        '''simple docstring'''
        while True:
               lowercase__:   Dict  			=   self.get_left_child_idx(lowerCAmelCase__ )  # noqa: E741
               lowercase__:   Any  			=   self.get_right_child_idx(lowerCAmelCase__ )

               lowercase__:   List[Any]  			=   idx
               if l < len(lowerCAmelCase__ ) and array[l] < array[idx]:
                      lowercase__:   List[str]  			=   l
               if r < len(lowerCAmelCase__ ) and array[r] < array[smallest]:
                      lowercase__:   Union[str, Any]  			=   r

               if smallest != idx:
                      lowercase__			,						lowercase__:   Tuple  			=   array[smallest], array[idx]
                      (
                          (
                          lowercase__
                      )			,						(
                          lowercase__
                      )			,						
                      ):   Union[str, Any]  			=   (
                          self.idx_of_element[array[smallest]],
                          self.idx_of_element[array[idx]],
                      )
                      lowercase__:   Any  			=   smallest
               else:
                      break
 def 		SCREAMING_SNAKE_CASE__   (   self				, lowerCAmelCase__ )					->				int:




        '''simple docstring'''
        lowercase__:   Tuple  			=   self.get_parent_idx(lowerCAmelCase__ )
        while p >= 0 and self.heap[p] > self.heap[idx]:
               lowercase__			,						lowercase__:   str  			=   self.heap[idx], self.heap[p]
               lowercase__			,						lowercase__:   Dict  			=   (
                   self.idx_of_element[self.heap[idx]],
                   self.idx_of_element[self.heap[p]],
               )
               lowercase__:   Any  			=   p
               lowercase__:   str  			=   self.get_parent_idx(lowerCAmelCase__ )
 def 		SCREAMING_SNAKE_CASE__   (   self )					->				List[Any]:




        '''simple docstring'''
        return self.heap[0]
 def 		SCREAMING_SNAKE_CASE__   (   self )					->				Tuple:




        '''simple docstring'''
        lowercase__			,						lowercase__:   Dict  			=   self.heap[-1], self.heap[0]
        lowercase__			,						lowercase__:   Union[str, Any]  			=   (
            self.idx_of_element[self.heap[-1]],
            self.idx_of_element[self.heap[0]],
        )

        lowercase__:   Union[str, Any]  			=   self.heap.pop()
        del self.idx_of_element[x]
        self.sift_down(0				, self.heap )
        return x
 def 		SCREAMING_SNAKE_CASE__   (   self				, lowerCAmelCase__ )					->				Any:




        '''simple docstring'''
        self.heap.append(lowerCAmelCase__ )
        lowercase__:   Tuple  			=   len(self.heap ) - 1
        lowercase__:   Tuple  			=   node.val
        self.sift_up(len(self.heap ) - 1 )
 def 		SCREAMING_SNAKE_CASE__   (   self )					->				int:




        '''simple docstring'''
        return len(self.heap ) == 0
 def 		SCREAMING_SNAKE_CASE__   (   self				, lowerCAmelCase__				, lowerCAmelCase__ )					->				Dict:




        '''simple docstring'''
        assert (
            self.heap[self.idx_of_element[node]].val > new_value
        ), "newValue must be less that current value"
        lowercase__:   Dict  			=   new_value
        lowercase__:   Tuple  			=   new_value
        self.sift_up(self.idx_of_element[node] )





__lowerCAmelCase		       =							Node('''R''', -1)
__lowerCAmelCase		       =							Node('''B''', 6)
__lowerCAmelCase		       =							Node('''A''', 3)
__lowerCAmelCase		       =							Node('''X''', 1)
__lowerCAmelCase		       =							Node('''E''', 4)

# Use one of these two ways to generate Min-Heap

# Generating Min-Heap from array
__lowerCAmelCase		       =							MinHeap([r, b, a, x, e])

# Generating Min-Heap by Insert method
# myMinHeap.insert(a)
# myMinHeap.insert(b)
# myMinHeap.insert(x)
# myMinHeap.insert(r)
# myMinHeap.insert(e)

# Before
print('''Min Heap - before decrease key''')
for i in my_min_heap.heap:
 print(i)

print('''Min Heap - After decrease key of node [B -> -17]''')
my_min_heap.decrease_key(b, -17)

# After
for i in my_min_heap.heap:
 print(i)

if __name__ == "__main__":
 import doctest

 doctest.testmod()






from __future__ import annotations

from collections.abc import Callable

__lowerCAmelCase		       =							list[list[float | int]]




def    snake_case_     (  snake_case			,							snake_case     )  ->				Matrix:
       lowercase__:   int  			=   len(snake_case     )
       lowercase__:   Matrix  			=   [[0 for _ in range(size + 1     )] for _ in range(snake_case     )]
       lowercase__:   int
       lowercase__:   int
       lowercase__:   int
       lowercase__:   int
       lowercase__:   int
       lowercase__:   float

       for row in range(snake_case     ):
              for col in range(snake_case     ):
                     lowercase__:   List[Any]  			=   matrix[row][col]

              lowercase__:   Optional[int]  			=   vector[row][0]

       lowercase__:   str  			=   0
       lowercase__:   Any  			=   0
       while row < size and col < size:
              # pivoting
              lowercase__:   List[Any]  			=   max((abs(augmented[rowa][col]     ), rowa) for rowa in range(snake_case			,							snake_case     )     )[
                  1
              ]
              if augmented[pivot_row][col] == 0:
                     col += 1
                     continue
              else:
                     lowercase__			,						lowercase__:   Optional[int]  			=   augmented[pivot_row], augmented[row]

              for rowa in range(row + 1			,							snake_case     ):
                     lowercase__:   Any  			=   augmented[rowa][col] / augmented[row][col]
                     lowercase__:   int  			=   0
                     for cola in range(col + 1			,							size + 1     ):
                            augmented[rowa][cola] -= augmented[row][cola] * ratio

              row += 1
              col += 1

       # back substitution
       for col in range(1			,							snake_case     ):
              for row in range(snake_case     ):
                     lowercase__:   Union[str, Any]  			=   augmented[row][col] / augmented[col][col]
                     for cola in range(snake_case			,							size + 1     ):
                            augmented[row][cola] -= augmented[col][cola] * ratio

    # round to get rid of numbers like 2.000000000000004
       return [
           [round(augmented[row][size] / augmented[row][row]			,							10     )] for row in range(snake_case     )
       ]




def    snake_case_     (  snake_case     )  ->				Callable[[int], int]:
       lowercase__:   int  			=   len(snake_case     )
       lowercase__:   Matrix  			=   [[0 for _ in range(snake_case     )] for _ in range(snake_case     )]
       lowercase__:   Matrix  			=   [[0] for _ in range(snake_case     )]
       lowercase__:   Matrix
       lowercase__:   int
       lowercase__:   int
       lowercase__:   int

       for x_val, y_val in enumerate(snake_case     ):
              for col in range(snake_case     ):
                     lowercase__:   List[str]  			=   (x_val + 1) ** (size - col - 1)
              lowercase__:   str  			=   y_val

       lowercase__:   Optional[int]  			=   solve(snake_case			,							snake_case     )

       def interpolated_func(snake_case     ) -> int:
              return sum(
                  round(coeffs[x_val][0]     ) * (var ** (size - x_val - 1))
                  for x_val in range(snake_case     )     )

       return interpolated_func




def    snake_case_     (  snake_case     )  ->				int:
       return (
           1
           - variable
           + variable**2
           - variable**3
           + variable**4
           - variable**5
           + variable**6
           - variable**7
           + variable**8
           - variable**9
           + variable**10
       )




def    snake_case_     (  snake_case = question_function			,							snake_case = 10     )  ->				int:
       lowercase__:   list[int]  			=   [func(snake_case     ) for x_val in range(1			,							order + 1     )]

       lowercase__:   list[Callable[[int], int]]  			=   [
           interpolate(data_points[:max_coeff]     ) for max_coeff in range(1			,							order + 1     )
       ]

       lowercase__:   int  			=   0
       lowercase__:   Callable[[int], int]
       lowercase__:   int

       for poly in polynomials:
              lowercase__:   List[str]  			=   1
              while func(snake_case     ) == poly(snake_case     ):
                     x_val += 1

              ret += poly(snake_case     )

       return ret


if __name__ == "__main__":
 print(F'''{solution() = }''')


