from __future__ import annotations

from random import random
from typing import Generic, TypeVar

__lowercase										= TypeVar('''KT''')
__lowercase										= TypeVar('''VT''')


class      lowerCamelCase_					(       Generic[KT, VT]			):






    '''simple docstring'''






    def __init__(     self			,			__lowercase = "root"			,			__lowercase = None)	->     List[str]:
        __UpperCamelCase					:Any        =		key
        __UpperCamelCase					:Optional[int]        =		value
        __UpperCamelCase					:list[Node[KT, VT]]        =		[]







    def __repr__(     self)	->     str:
        return f"""Node({self.key}: {self.value})"""







    @property
    def       UpperCamelCase__       (     self)	->     int:
        return len(self.forward)


class      lowerCamelCase_					(       Generic[KT, VT]			):






    '''simple docstring'''






    def __init__(     self			,			__lowercase = 0.5			,			__lowercase = 16)	->     Tuple:
        __UpperCamelCase					:Node[KT, VT]        =		Node[KT, VT]()
        __UpperCamelCase					:str        =		0
        __UpperCamelCase					:List[Any]        =		p
        __UpperCamelCase					:Optional[int]        =		max_level







    def __str__(     self)	->     str:
        __UpperCamelCase					:List[Any]        =		list(self)

        if len(__lowercase) == 0:
            return f"""SkipList(level={self.level})"""

        __UpperCamelCase					:List[str]        =		max((len(str(__lowercase)) for item in items)			,			default=4)
        __UpperCamelCase					:Union[str, Any]        =		max(__lowercase			,			4) + 4

        __UpperCamelCase					:Optional[int]        =		self.head
        __UpperCamelCase					:Optional[Any]        =		[]

        __UpperCamelCase					:List[str]        =		node.forward.copy()
        lines.append(f"""[{node.key}]""".ljust(__lowercase			,			'''-''') + '''* ''' * len(__lowercase))
        lines.append(''' ''' * label_size + '''| ''' * len(__lowercase))

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

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

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







    def __iter__(     self)	->     Union[str, Any]:
        __UpperCamelCase					:Optional[int]        =		self.head

        while len(node.forward) != 0:
            yield node.forward[0].key
            __UpperCamelCase					:Optional[Any]        =		node.forward[0]







    def       UpperCamelCase__       (     self)	->     int:
        __UpperCamelCase					:Any        =		1
        while random() < self.p and level < self.max_level:
            level += 1

        return level







    def       UpperCamelCase__       (     self			,			__lowercase)	->     tuple[Node[KT, VT] | None, list[Node[KT, VT]]]:
        __UpperCamelCase					:List[Any]        =		[]

        __UpperCamelCase					:Dict        =		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:
                __UpperCamelCase					:int        =		node.forward[i]
            # Each leftmost node (relative to searched node) will potentially have to
            # be updated.
            update_vector.append(__lowercase)

        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			,			__lowercase)	->     Optional[Any]:
        __UpperCamelCase  ,	__UpperCamelCase					:Optional[int]        =		self._locate_node(__lowercase)

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







    def       UpperCamelCase__       (     self			,			__lowercase			,			__lowercase)	->     List[str]:
        __UpperCamelCase  ,	__UpperCamelCase					:Dict        =		self._locate_node(__lowercase)
        if node is not None:
            __UpperCamelCase					:Tuple        =		value
        else:
            __UpperCamelCase					:Union[str, Any]        =		self.random_level()

            if level > self.level:
                # After level increase we have to add additional nodes to head.
                for _ in range(self.level - 1			,			__lowercase):
                    update_vector.append(self.head)
                __UpperCamelCase					:Optional[int]        =		level

            __UpperCamelCase					:List[str]        =		Node(__lowercase			,			__lowercase)

            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(__lowercase)
                else:
                    __UpperCamelCase					:Optional[int]        =		new_node







    def       UpperCamelCase__       (     self			,			__lowercase)	->     VT | None:
        __UpperCamelCase  ,	__UpperCamelCase					:int        =		self._locate_node(__lowercase)

        if node is not None:
            return node.value

        return None



def lowerCamelCase  (       ):
    '''simple docstring'''







    __UpperCamelCase					:List[str]        =		SkipList()
    skip_list.insert('''Key1'''  ,      3     )
    skip_list.insert('''Key2'''  ,      12     )
    skip_list.insert('''Key3'''  ,      41     )
    skip_list.insert('''Key4'''  ,      -19     )

    __UpperCamelCase					:Optional[Any]        =		skip_list.head
    __UpperCamelCase					:List[str]        =		{}
    while node.level != 0:
        __UpperCamelCase					:str        =		node.forward[0]
        __UpperCamelCase					:Optional[int]        =		node.value

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



def lowerCamelCase  (       ):
    '''simple docstring'''







    __UpperCamelCase					:int        =		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     )

    __UpperCamelCase					:int        =		skip_list.head
    __UpperCamelCase					:Optional[Any]        =		{}
    while node.level != 0:
        __UpperCamelCase					:Union[str, Any]        =		node.forward[0]
        __UpperCamelCase					:Any        =		node.value

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



def lowerCamelCase  (       ):
    '''simple docstring'''







    __UpperCamelCase					:Any        =		SkipList()
    assert skip_list.find('''Some key'''     ) is None



def lowerCamelCase  (       ):
    '''simple docstring'''







    __UpperCamelCase					:List[Any]        =		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 lowerCamelCase  (       ):
    '''simple docstring'''







    __UpperCamelCase					:Union[str, Any]        =		SkipList()
    skip_list.delete('''Some key'''     )

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



def lowerCamelCase  (       ):
    '''simple docstring'''







    __UpperCamelCase					:List[str]        =		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 lowerCamelCase  (       ):
    '''simple docstring'''







    __UpperCamelCase					:str        =		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 lowerCamelCase  (       ):
    '''simple docstring'''







    __UpperCamelCase					:Union[str, Any]        =		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(SCREAMING_SNAKE_CASE     ):
        yield node.key
        for forward_node in node.forward:
            yield from traverse_keys(SCREAMING_SNAKE_CASE     )

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



def lowerCamelCase  (       ):
    '''simple docstring'''







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

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



def lowerCamelCase  (       ):
    '''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 lowerCamelCase  (       ):
    '''simple docstring'''







    __UpperCamelCase					:Union[str, Any]        =		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(SCREAMING_SNAKE_CASE     )


if __name__ == "__main__":
    import doctest

    doctest.testmod()
    main()






def lowerCamelCase  (       SCREAMING_SNAKE_CASE     ):
    '''simple docstring'''







    if not grid or not grid[0]:
        raise TypeError('''The grid does not contain the appropriate information'''     )

    for cell_n in range(1  ,      len(grid[0]     )     ):
        grid[0][cell_n] += grid[0][cell_n - 1]
    __UpperCamelCase					:Union[str, Any]        =		grid[0]

    for row_n in range(1  ,      len(SCREAMING_SNAKE_CASE     )     ):
        __UpperCamelCase					:Optional[int]        =		grid[row_n]
        __UpperCamelCase					:Dict        =		fill_row(SCREAMING_SNAKE_CASE  ,      SCREAMING_SNAKE_CASE     )
        __UpperCamelCase					:Any        =		grid[row_n]

    return grid[-1][-1]



def lowerCamelCase  (       SCREAMING_SNAKE_CASE  ,      SCREAMING_SNAKE_CASE     ):
    '''simple docstring'''







    current_row[0] += row_above[0]
    for cell_n in range(1  ,      len(SCREAMING_SNAKE_CASE     )     ):
        current_row[cell_n] += min(current_row[cell_n - 1]  ,      row_above[cell_n]     )

    return current_row


if __name__ == "__main__":
    import doctest

    doctest.testmod()





