

"""simple docstring"""






import random

def 		lowerCamelCase__		(					_lowerCamelCase ,   _lowerCamelCase ,   _lowerCamelCase = False   ):

    '''simple docstring'''



    _lowerCAmelCase       :   dict  =    {i: [] for i in range(_lowerCamelCase   )}

    # if probability is greater or equal than 1, then generate a complete graph
    if probability >= 1:
        return complete_graph(_lowerCamelCase   )
    # if probability is lower or equal than 0, then return a graph without edges
    if probability <= 0:
        return graph

    # for each couple of nodes, add an edge from u to v
    # if the number randomly generated is greater than probability probability
    for i in range(_lowerCamelCase   ):
        for j in range(i + 1 ,   _lowerCamelCase   ):
            if random.random() < probability:
                graph[i].append(_lowerCamelCase   )
                if not directed:
                    # if the graph is undirected, add an edge in from j to i, either
                    graph[j].append(_lowerCamelCase   )
    return graph

def 		lowerCamelCase__		(					_lowerCamelCase   ):

    '''simple docstring'''



    return {
        i: [j for j in range(_lowerCamelCase   ) if i != j] for i in range(_lowerCamelCase   )
    }


if __name__ == "__main__":
 import doctest

 doctest.testmod()








"""simple docstring"""






from __future__ import annotations

from collections.abc import Callable

_lowerCAmelCase						=		list[list[float | int]]

def 		lowerCamelCase__		(					_lowerCamelCase ,   _lowerCamelCase   ):

    '''simple docstring'''



    _lowerCAmelCase       :   int  =    len(_lowerCamelCase   )
    _lowerCAmelCase       :   Matrix  =    [[0 for _ in range(size + 1   )] for _ in range(_lowerCamelCase   )]
    _lowerCAmelCase       :   int
    _lowerCAmelCase       :   int
    _lowerCAmelCase       :   int
    _lowerCAmelCase       :   int
    _lowerCAmelCase       :   int
    _lowerCAmelCase       :   float

    for row in range(_lowerCamelCase   ):
        for col in range(_lowerCamelCase   ):
            _lowerCAmelCase       :   Union[str, Any]  =    matrix[row][col]

        _lowerCAmelCase       :   Tuple  =    vector[row][0]

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

        for rowa in range(row + 1 ,   _lowerCamelCase   ):
            _lowerCAmelCase       :   Dict  =    augmented[rowa][col] / augmented[row][col]
            _lowerCAmelCase       :   Optional[Any]  =    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 ,   _lowerCamelCase   ):
        for row in range(_lowerCamelCase   ):
            _lowerCAmelCase       :   int  =    augmented[row][col] / augmented[col][col]
            for cola in range(_lowerCamelCase ,   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(_lowerCamelCase   )
    ]

def 		lowerCamelCase__		(					_lowerCamelCase   ):

    '''simple docstring'''



    _lowerCAmelCase       :   int  =    len(_lowerCamelCase   )
    _lowerCAmelCase       :   Matrix  =    [[0 for _ in range(_lowerCamelCase   )] for _ in range(_lowerCamelCase   )]
    _lowerCAmelCase       :   Matrix  =    [[0] for _ in range(_lowerCamelCase   )]
    _lowerCAmelCase       :   Matrix
    _lowerCAmelCase       :   int
    _lowerCAmelCase       :   int
    _lowerCAmelCase       :   int

    for x_val, y_val in enumerate(_lowerCamelCase   ):
        for col in range(_lowerCamelCase   ):
            _lowerCAmelCase       :   Union[str, Any]  =    (x_val + 1) ** (size - col - 1)
        _lowerCAmelCase       :   Optional[int]  =    y_val

    _lowerCAmelCase       :   List[Any]  =    solve(_lowerCamelCase ,   _lowerCamelCase   )

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

    return interpolated_func

def 		lowerCamelCase__		(					_lowerCamelCase   ):

    '''simple docstring'''



    return (
        1
        - variable
        + variable**2
        - variable**3
        + variable**4
        - variable**5
        + variable**6
        - variable**7
        + variable**8
        - variable**9
        + variable**10
    )

def 		lowerCamelCase__		(					_lowerCamelCase = question_function ,   _lowerCamelCase = 10   ):

    '''simple docstring'''



    _lowerCAmelCase       :   list[int]  =    [func(_lowerCamelCase   ) for x_val in range(1 ,   order + 1   )]

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

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

    for poly in polynomials:
        _lowerCAmelCase       :   Any  =    1
        while func(_lowerCamelCase   ) == poly(_lowerCamelCase   ):
            x_val += 1

        ret += poly(_lowerCamelCase   )

    return ret


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





