
from __future__ import annotations

from random import random
from typing import Generic, TypeVar

_UpperCAmelCase					: Tuple      =       TypeVar("KT")
_UpperCAmelCase					: Union[str, Any]      =       TypeVar("VT")


class 				lowercase					(		Generic[KT, VT]   ):



		def __init__( self				,      A_ = "root"				,      A_ = None       )  ->					int:
				"""simple docstring"""
				UpperCamelCase		=     key
				UpperCamelCase		=     value
				UpperCamelCase		=     []



		def __repr__( self       )  ->					str:
				"""simple docstring"""
				return F'''Node({self.key}: {self.value})'''



		@property
		def __UpperCamelCase ( self       )  ->					int:
				"""simple docstring"""
				return len(self.forward       )






class 				lowercase					(		Generic[KT, VT]   ):



		def __init__( self				,      A_ = 0.5				,      A_ = 16       )  ->					int:
				"""simple docstring"""
				UpperCamelCase		=     Node[KT, VT]()
				UpperCamelCase		=     0
				UpperCamelCase		=     p
				UpperCamelCase		=     max_level



		def __str__( self       )  ->					str:
				"""simple docstring"""
				UpperCamelCase		=     list(self       )

				if len(A_       ) == 0:
						return F'''SkipList(level={self.level})'''

				UpperCamelCase		=     max((len(str(A_       )       ) for item in items)				,      default=4       )
				UpperCamelCase		=     max(A_				,      4       ) + 4

				UpperCamelCase		=     self.head
				UpperCamelCase		=     []

				UpperCamelCase		=     node.forward.copy()
				lines.append(F'''[{node.key}]'''.ljust(A_				,      '-'       ) + '* ' * len(A_       )       )
				lines.append(' ' * label_size + '| ' * len(A_       )       )

				while len(node.forward       ) != 0:
						UpperCamelCase		=     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_       )       )
						UpperCamelCase		=     node.forward

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



		def __iter__( self       )  ->					Any:
				"""simple docstring"""
				UpperCamelCase		=     self.head

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



		def __UpperCamelCase ( self       )  ->					int:
				"""simple docstring"""
				UpperCamelCase		=     1
				while random() < self.p and level < self.max_level:
						level += 1

				return level



		def __UpperCamelCase ( self				,      A_       )  ->					tuple[Node[KT, VT] | None, list[Node[KT, VT]]]:
				"""simple docstring"""
				UpperCamelCase		=     []

				UpperCamelCase		=     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		=     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				,      A_       )  ->					str:
				"""simple docstring"""
				UpperCamelCase	,  UpperCamelCase		=     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:
												UpperCamelCase		=     node.forward[i]
										else:
												UpperCamelCase		=     update_node.forward[:i]



		def __UpperCamelCase ( self				,      A_				,      A_       )  ->					List[str]:
				"""simple docstring"""
				UpperCamelCase	,  UpperCamelCase		=     self._locate_node(A_       )
				if node is not None:
						UpperCamelCase		=     value
				else:
						UpperCamelCase		=     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       )
								UpperCamelCase		=     level

						UpperCamelCase		=     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:
										UpperCamelCase		=     new_node



		def __UpperCamelCase ( self				,      A_       )  ->					VT | None:
				"""simple docstring"""
				UpperCamelCase	,  UpperCamelCase		=     self._locate_node(A_       )

				if node is not None:
						return node.value

				return None






def       A       (					)   ->  Any:
		'''simple docstring'''

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

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

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


def       A       (					)   ->  Union[str, Any]:
		'''simple docstring'''

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

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


def       A       (					)   ->  Optional[Any]:
		'''simple docstring'''

		UpperCamelCase		=     SkipList()
		assert skip_list.find('Some key' ) is None


def       A       (					)   ->  Union[str, Any]:
		'''simple docstring'''

		UpperCamelCase		=     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       A       (					)   ->  int:
		'''simple docstring'''

		UpperCamelCase		=     SkipList()
		skip_list.delete('Some key' )

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


def       A       (					)   ->  Tuple:
		'''simple docstring'''

		UpperCamelCase		=     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       A       (					)   ->  List[str]:
		'''simple docstring'''

		UpperCamelCase		=     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       A       (					)   ->  int:
		'''simple docstring'''

		UpperCamelCase		=     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(lowercase ):
				yield node.key
				for forward_node in node.forward:
						yield from traverse_keys(lowercase )

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


def       A       (					)   ->  Optional[Any]:
		'''simple docstring'''

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

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


def       A       (					)   ->  Union[str, 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       A       (					)   ->  int:
		'''simple docstring'''

		UpperCamelCase		=     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(lowercase )


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					main()


from random import randint
from tempfile import TemporaryFile

import numpy as np


def       A       (					lowercase    ,       lowercase    ,       lowercase )   ->  str:
		'''simple docstring'''

		UpperCamelCase		=     0
		if start < end:
				UpperCamelCase		=     randint(lowercase    ,       lowercase )
				UpperCamelCase		=     a[end]
				UpperCamelCase		=     a[pivot]
				UpperCamelCase		=     temp

				UpperCamelCase	,  UpperCamelCase		=     _in_place_partition(lowercase    ,       lowercase    ,       lowercase )
				count += _in_place_quick_sort(lowercase    ,       lowercase    ,       p - 1 )
				count += _in_place_quick_sort(lowercase    ,       p + 1    ,       lowercase )
		return count


def       A       (					lowercase    ,       lowercase    ,       lowercase )   ->  int:
		'''simple docstring'''

		UpperCamelCase		=     0
		UpperCamelCase		=     randint(lowercase    ,       lowercase )
		UpperCamelCase		=     a[end]
		UpperCamelCase		=     a[pivot]
		UpperCamelCase		=     temp
		UpperCamelCase		=     start - 1
		for index in range(lowercase    ,       lowercase ):
				count += 1
				if a[index] < a[end]:  # check if current val is less than pivot value
						UpperCamelCase		=     new_pivot_index + 1
						UpperCamelCase		=     a[new_pivot_index]
						UpperCamelCase		=     a[index]
						UpperCamelCase		=     temp

		UpperCamelCase		=     a[new_pivot_index + 1]
		UpperCamelCase		=     a[end]
		UpperCamelCase		=     temp
		return new_pivot_index + 1, count


_UpperCAmelCase					: Union[str, Any]      =       TemporaryFile()
_UpperCAmelCase					: List[Any]      =       100  # 1000 elements are to be sorted


_UpperCAmelCase   ,_UpperCAmelCase					: Any      =       0, 1  # mean and standard deviation
_UpperCAmelCase					: Any      =       np.random.normal(mu, sigma, p)
np.save(outfile, X)
print("The array is")
print(X)


outfile.seek(0)  # using the same array
_UpperCAmelCase					: Any      =       np.load(outfile)
_UpperCAmelCase					: str      =       len(M) - 1
_UpperCAmelCase					: List[str]      =       _in_place_quick_sort(M, 0, r)

print(
    "No of Comparisons for 100 elements selected from a standard normal distribution"
    "is :"
)


print(z)
