
from __future__ import annotations

from random import random
from typing import Generic, TypeVar

_A   =			TypeVar("KT")
_A   =			TypeVar("VT")







class 					lowerCamelCase		(  Generic[KT, VT]  ):





     def __init__(self				:   Union[str, Any]    ,   _A				:   KT | str = "root"    ,   _A				:   VT | None = None       )						->						Union[str, Any]:
         snake_case								= key
         snake_case								= value
         snake_case								= []





     def __repr__(self				:   Union[str, Any]       )						->						Any:
         return f'Node({self.key}: {self.value})'





     @property
     def 		UpperCAmelCase(self				:   str       )						->						Any:
         return len(self.forward       )







class 					lowerCamelCase		(  Generic[KT, VT]  ):





     def __init__(self				:   str    ,   _A				:   float = 0.5    ,   _A				:   int = 1_6       )						->						Optional[Any]:
         snake_case								= Node[KT, VT]()
         snake_case								= 0
         snake_case								= p
         snake_case								= max_level





     def __str__(self				:   Optional[Any]       )						->						Union[str, Any]:
         snake_case								= list(self       )

         if len(__A       ) == 0:
             return f'SkipList(level={self.level})'

         snake_case								= max((len(str(__A       )       ) for item in items)    ,   default=4       )
         snake_case								= max(__A    ,   4       ) + 4

         snake_case								= self.head
         snake_case								= []

         snake_case								= node.forward.copy()
         lines.append(f'[{node.key}]'.ljust(__A    ,   "-"       ) + "* " * len(__A       )       )
         lines.append(" " * label_size + "| " * len(__A       )       )

         while len(node.forward       ) != 0:
             snake_case								= node.forward[0]

             lines.append(
                 f'[{node.key}]'.ljust(__A    ,   "-"       )
                 + " ".join(str(n.key       ) if n.key == node.key else "|" for n in forwards       )       )
             lines.append(" " * label_size + "| " * len(__A       )       )
             snake_case								= node.forward

         lines.append("None".ljust(__A       ) + "* " * len(__A       )       )
         return f'SkipList(level={self.level})\n' + "\n".join(__A       )





     def __iter__(self				:   Dict       )						->						Optional[int]:
         snake_case								= self.head

         while len(node.forward       ) != 0:
             yield node.forward[0].key
             snake_case								= node.forward[0]





     def 		UpperCAmelCase(self				:   str       )						->						Tuple:
         snake_case								= 1
         while random() < self.p and level < self.max_level:
             level += 1

         return level





     def 		UpperCAmelCase(self				:   Union[str, Any]    ,   _A				:   List[str]       )						->						Optional[int]:
         snake_case								= []

         snake_case								= self.head

         for i in reversed(range(self.level       )       ):
             # i < node.level - When node level is lesser than `i` decrement `i`.
             # node.forward[i].key < key - Jumping to node with key value higher
             #                             or equal to searched key would result
             #                             in skipping searched key.
             while i < node.level and node.forward[i].key < key:
                 snake_case								= node.forward[i]
             # Each leftmost node (relative to searched node) will potentially have to
             # be updated.
             update_vector.append(__A       )

         update_vector.reverse()  # Note that we were inserting values in reverse order.

         # len(node.forward) != 0 - If current node doesn't contain any further
         #                          references then searched key is not present.
         # node.forward[0].key == key - Next node key should be equal to search key
         #                              if key is present.
         if len(node.forward       ) != 0 and node.forward[0].key == key:
             return node.forward[0], update_vector
         else:
             return None, update_vector





     def 		UpperCAmelCase(self				:   Tuple    ,   _A				:   KT       )						->						List[Any]:
         snake_case								= self._locate_node(__A       )

         if node is not None:
             for i, update_node in enumerate(__A       ):
                 # Remove or replace all references to removed node.
                 if update_node.level > i and update_node.forward[i].key == key:
                     if node.level > i:
                         snake_case								= node.forward[i]
                     else:
                         snake_case								= update_node.forward[:i]





     def 		UpperCAmelCase(self				:   int    ,   _A				:   KT    ,   _A				:   VT       )						->						Optional[int]:
         snake_case								= self._locate_node(__A       )
         if node is not None:
             snake_case								= value
         else:
             snake_case								= self.random_level()

             if level > self.level:
                 # After level increase we have to add additional nodes to head.
                 for _ in range(self.level - 1    ,   __A       ):
                     update_vector.append(self.head       )
                 snake_case								= level

             snake_case								= Node(__A    ,   __A       )

             for i, update_node in enumerate(update_vector[:level]       ):
                 # Change references to pass through new node.
                 if update_node.level > i:
                     new_node.forward.append(update_node.forward[i]       )

                 if update_node.level < i + 1:
                     update_node.forward.append(__A       )
                 else:
                     snake_case								= new_node





     def 		UpperCAmelCase(self				:   Tuple    ,   _A				:   VT       )						->						Tuple:
         snake_case								= self._locate_node(__A       )

         if node is not None:
             return node.value

         return None



def 						lowercase_      (       )							->							List[Any]:
    """simple docstring"""

    snake_case								= SkipList()
    skip_list.insert("Key1"			,		3		)
    skip_list.insert("Key2"			,		12		)
    skip_list.insert("Key3"			,		41		)
    skip_list.insert("Key4"			,		-19		)

    snake_case								= skip_list.head
    snake_case								= {}
    while node.level != 0:
        snake_case								= node.forward[0]
        snake_case								= node.value

    assert len(A__		) == 4
    assert all_values["Key1"] == 3
    assert all_values["Key2"] == 12
    assert all_values["Key3"] == 41
    assert all_values["Key4"] == -19



def 						lowercase_      (       )							->							Dict:
    """simple docstring"""

    snake_case								= SkipList()
    skip_list.insert("Key1"			,		10		)
    skip_list.insert("Key1"			,		12		)

    skip_list.insert("Key5"			,		7		)
    skip_list.insert("Key7"			,		10		)
    skip_list.insert("Key10"			,		5		)

    skip_list.insert("Key7"			,		7		)
    skip_list.insert("Key5"			,		5		)
    skip_list.insert("Key10"			,		10		)

    snake_case								= skip_list.head
    snake_case								= {}
    while node.level != 0:
        snake_case								= node.forward[0]
        snake_case								= node.value

    if len(A__		) != 4:
        print()
    assert len(A__		) == 4
    assert all_values["Key1"] == 12
    assert all_values["Key7"] == 7
    assert all_values["Key5"] == 5
    assert all_values["Key10"] == 10



def 						lowercase_      (       )							->							Any:
    """simple docstring"""

    snake_case								= SkipList()
    assert skip_list.find("Some key"		) is None



def 						lowercase_      (       )							->							Union[str, Any]:
    """simple docstring"""

    snake_case								= SkipList()

    skip_list.insert("Key2"			,		20		)
    assert skip_list.find("Key2"		) == 20

    skip_list.insert("Some Key"			,		10		)
    skip_list.insert("Key2"			,		8		)
    skip_list.insert("V"			,		13		)

    assert skip_list.find("Y"		) is None
    assert skip_list.find("Key2"		) == 8
    assert skip_list.find("Some Key"		) == 10
    assert skip_list.find("V"		) == 13



def 						lowercase_      (       )							->							List[Any]:
    """simple docstring"""

    snake_case								= SkipList()
    skip_list.delete("Some key"		)

    assert len(skip_list.head.forward		) == 0



def 						lowercase_      (       )							->							Optional[Any]:
    """simple docstring"""

    snake_case								= SkipList()

    skip_list.insert("Key1"			,		12		)
    skip_list.insert("V"			,		13		)
    skip_list.insert("X"			,		14		)
    skip_list.insert("Key2"			,		15		)

    skip_list.delete("V"		)
    skip_list.delete("Key2"		)

    assert skip_list.find("V"		) is None
    assert skip_list.find("Key2"		) is None



def 						lowercase_      (       )							->							str:
    """simple docstring"""

    snake_case								= SkipList()

    skip_list.insert("Key1"			,		12		)
    skip_list.insert("V"			,		13		)
    skip_list.insert("X"			,		14		)
    skip_list.insert("Key2"			,		15		)

    skip_list.delete("V"		)
    assert skip_list.find("V"		) is None
    assert skip_list.find("X"		) == 14
    assert skip_list.find("Key1"		) == 12
    assert skip_list.find("Key2"		) == 15

    skip_list.delete("X"		)
    assert skip_list.find("V"		) is None
    assert skip_list.find("X"		) is None
    assert skip_list.find("Key1"		) == 12
    assert skip_list.find("Key2"		) == 15

    skip_list.delete("Key1"		)
    assert skip_list.find("V"		) is None
    assert skip_list.find("X"		) is None
    assert skip_list.find("Key1"		) is None
    assert skip_list.find("Key2"		) == 15

    skip_list.delete("Key2"		)
    assert skip_list.find("V"		) is None
    assert skip_list.find("X"		) is None
    assert skip_list.find("Key1"		) is None
    assert skip_list.find("Key2"		) is None



def 						lowercase_      (       )							->							Union[str, Any]:
    """simple docstring"""

    snake_case								= SkipList()

    skip_list.insert("Key1"			,		12		)
    skip_list.insert("V"			,		13		)
    skip_list.insert("X"			,		142		)
    skip_list.insert("Key2"			,		15		)

    skip_list.delete("X"		)

    def traverse_keys(A__		):
        yield node.key
        for forward_node in node.forward:
            yield from traverse_keys(A__		)

    assert len(set(traverse_keys(skip_list.head		)		)		) == 4



def 						lowercase_      (       )							->							List[str]:
    """simple docstring"""

    def is_sorted(A__		):
        return all(next_item >= item for item, next_item in zip(A__			,		lst[1:]		)		)

    snake_case								= SkipList()
    for i in range(10		):
        skip_list.insert(A__			,		A__		)
    assert is_sorted(list(A__		)		)
    skip_list.delete(5		)
    skip_list.delete(8		)
    skip_list.delete(2		)
    assert is_sorted(list(A__		)		)
    skip_list.insert(-12			,		-12		)
    skip_list.insert(77			,		77		)
    assert is_sorted(list(A__		)		)



def 						lowercase_      (       )							->							Any:
    """simple docstring"""

    for _ in range(100		):
        # Repeat test 100 times due to the probabilistic nature of skip list
        # random values == random bugs
        test_insert()
        test_insert_overrides_existing_value()

        test_searching_empty_list_returns_none()
        test_search()

        test_deleting_item_from_empty_list_do_nothing()
        test_deleted_items_are_not_founded_by_find_method()
        test_delete_removes_only_given_key()
        test_delete_doesnt_leave_dead_nodes()

        test_iter_always_yields_sorted_values()



def 						lowercase_      (       )							->							Dict:
    """simple docstring"""

    snake_case								= SkipList()
    skip_list.insert(2			,		"2"		)
    skip_list.insert(4			,		"4"		)
    skip_list.insert(6			,		"4"		)
    skip_list.insert(4			,		"5"		)
    skip_list.insert(8			,		"4"		)
    skip_list.insert(9			,		"4"		)

    skip_list.delete(4		)

    print(A__		)


if __name__ == "__main__":
  import doctest

  doctest.testmod()
  main()



from __future__ import annotations

from decimal import Decimal

from numpy import array



def 						lowercase_      (       A__		)							->							list[list[float]]:
    """simple docstring"""

    snake_case								= Decimal

    # Check if the provided matrix has 2 rows and 2 columns
    # since this implementation only works for 2x2 matrices
    if len(A__		) == 2 and len(matrix[0]		) == 2 and len(matrix[1]		) == 2:
        # Calculate the determinant of the matrix
        snake_case								= float(
            d(matrix[0][0]		) * d(matrix[1][1]		) - d(matrix[1][0]		) * d(matrix[0][1]		)		)
        if determinant == 0:
            raise ValueError("This matrix has no inverse."		)

        # Creates a copy of the matrix with swapped positions of the elements
        snake_case								= [[0.0, 0.0], [0.0, 0.0]]
        snake_case					,							snake_case								= matrix[1][1], matrix[0][0]
        snake_case					,							snake_case								= -matrix[1][0], -matrix[0][1]

        # Calculate the inverse of the matrix
        return [
            [(float(d(A__		)		) / determinant) or 0.0 for n in row] for row in swapped_matrix
        ]
    elif (
        len(A__		) == 3
        and len(matrix[0]		) == 3
        and len(matrix[1]		) == 3
        and len(matrix[2]		) == 3
    ):
        # Calculate the determinant of the matrix using Sarrus rule
        snake_case								= float(
            (
                (d(matrix[0][0]		) * d(matrix[1][1]		) * d(matrix[2][2]		))
                + (d(matrix[0][1]		) * d(matrix[1][2]		) * d(matrix[2][0]		))
                + (d(matrix[0][2]		) * d(matrix[1][0]		) * d(matrix[2][1]		))
            )
            - (
                (d(matrix[0][2]		) * d(matrix[1][1]		) * d(matrix[2][0]		))
                + (d(matrix[0][1]		) * d(matrix[1][0]		) * d(matrix[2][2]		))
                + (d(matrix[0][0]		) * d(matrix[1][2]		) * d(matrix[2][1]		))
            )		)
        if determinant == 0:
            raise ValueError("This matrix has no inverse."		)

        # Creating cofactor matrix
        snake_case								= [
            [d(0.0		), d(0.0		), d(0.0		)],
            [d(0.0		), d(0.0		), d(0.0		)],
            [d(0.0		), d(0.0		), d(0.0		)],
        ]
        snake_case								= (d(matrix[1][1]		) * d(matrix[2][2]		)) - (
            d(matrix[1][2]		) * d(matrix[2][1]		)
        )
        snake_case								= -(
            (d(matrix[1][0]		) * d(matrix[2][2]		)) - (d(matrix[1][2]		) * d(matrix[2][0]		))
        )
        snake_case								= (d(matrix[1][0]		) * d(matrix[2][1]		)) - (
            d(matrix[1][1]		) * d(matrix[2][0]		)
        )
        snake_case								= -(
            (d(matrix[0][1]		) * d(matrix[2][2]		)) - (d(matrix[0][2]		) * d(matrix[2][1]		))
        )
        snake_case								= (d(matrix[0][0]		) * d(matrix[2][2]		)) - (
            d(matrix[0][2]		) * d(matrix[2][0]		)
        )
        snake_case								= -(
            (d(matrix[0][0]		) * d(matrix[2][1]		)) - (d(matrix[0][1]		) * d(matrix[2][0]		))
        )
        snake_case								= (d(matrix[0][1]		) * d(matrix[1][2]		)) - (
            d(matrix[0][2]		) * d(matrix[1][1]		)
        )
        snake_case								= -(
            (d(matrix[0][0]		) * d(matrix[1][2]		)) - (d(matrix[0][2]		) * d(matrix[1][0]		))
        )
        snake_case								= (d(matrix[0][0]		) * d(matrix[1][1]		)) - (
            d(matrix[0][1]		) * d(matrix[1][0]		)
        )

        # Transpose the cofactor matrix (Adjoint matrix)
        snake_case								= array(A__		)
        for i in range(3		):
            for j in range(3		):
                snake_case								= cofactor_matrix[j][i]

        # Inverse of the matrix using the formula (1/determinant) * adjoint matrix
        snake_case								= array(A__		)
        for i in range(3		):
            for j in range(3		):
                inverse_matrix[i][j] /= d(A__		)

        # Calculate the inverse of the matrix
        return [[float(d(A__		)		) or 0.0 for n in row] for row in inverse_matrix]
    raise ValueError("Please provide a matrix of size 2x2 or 3x3."		)

