



def 	a						(				_UpperCAmelCase							: list[list[int]]			,       _UpperCAmelCase							: list[list[int]]						):







		'''simple docstring'''





		__UpperCAmelCase   :					Tuple													=     []
		__UpperCAmelCase   :					str													=     len(_UpperCAmelCase						)
		for i in range(_UpperCAmelCase						):
				__UpperCAmelCase   :					int													=     []
				for j in range(_UpperCAmelCase						):
						__UpperCAmelCase   :					Optional[Any]													=     0
						for k in range(_UpperCAmelCase						):
								__UpperCAmelCase   :					Optional[int]													=     val + matrix_a[i][k] * matrix_b[k][j]
						list_a.append(_UpperCAmelCase						)
				matrix_c.append(_UpperCAmelCase						)
		return matrix_c

def 	a						(				_UpperCAmelCase							: int						):







		'''simple docstring'''





		return [[int(row == column						) for column in range(_UpperCAmelCase						)] for row in range(_UpperCAmelCase						)]

def 	a						(				_UpperCAmelCase							: int						):







		'''simple docstring'''





		if n <= 1:
				return n
		__UpperCAmelCase   :					Optional[Any]													=     identity(2						)
		__UpperCAmelCase   :					List[Any]													=     [[1, 1], [1, 0]]
		__UpperCAmelCase   :					Dict													=     n - 1
		while n > 0:
				if n % 2 == 1:
						__UpperCAmelCase   :					Union[str, Any]													=     multiply(_UpperCAmelCase			,       _UpperCAmelCase						)
				__UpperCAmelCase   :					Optional[int]													=     multiply(_UpperCAmelCase			,       _UpperCAmelCase						)
				__UpperCAmelCase   :					List[str]													=     int(n / 2						)
		return res_matrix[0][0]

def 	a						(				_UpperCAmelCase							: int						):







		'''simple docstring'''





		if n <= 1:
				return n
		__UpperCAmelCase   :					Optional[int]													=     0
		__UpperCAmelCase   :					str													=     1
		for _ in range(2			,       n + 1						):
				__UpperCAmelCase							, __UpperCAmelCase   :					Optional[Any]													=     fiba, fiba + fiba
		return fiba

def 	a						(				):







		'''simple docstring'''





		for ordinal in "0th 1st 2nd 3rd 10th 100th 1000th".split():
				__UpperCAmelCase   :					Optional[Any]													=     int(''''''.join(c for c in ordinal if c in '''0123456789'''						)						)  # 1000th --> 1000
				print(
				    f'{ordinal} fibonacci number using matrix exponentiation is '
				    f'{nth_fibonacci_matrix(_UpperCAmelCase						)} and using bruteforce is '
				    f'{nth_fibonacci_bruteforce(_UpperCAmelCase						)}\n'						)
		# from timeit import timeit
		# print(timeit("nth_fibonacci_matrix(1000000)",
		#              "from main import nth_fibonacci_matrix", number=5))
		# print(timeit("nth_fibonacci_bruteforce(1000000)",
		#              "from main import nth_fibonacci_bruteforce", number=5))
		# 2.3342058970001744
		# 57.256506615000035


if __name__ == "__main__":
				import doctest

				doctest.testmod()
				main()










from __future__ import annotations

import random
from collections.abc import Iterable
from typing import Any, Generic, TypeVar

__A             =TypeVar("T", bound=bool)

class 		UpperCAmelCase__      (      Generic[T]       ):




			'''simple docstring'''






			def __init__(       self							:			List[Any]	,	a_							:			T			):
					'''simple docstring'''





					__UpperCAmelCase   :					T													=     value
					__UpperCAmelCase   :					RandomizedHeapNode[T] | None													=     None
					__UpperCAmelCase   :					RandomizedHeapNode[T] | None													=     None




			@property
			def 			snake_case__					(       self							:			Any			):
					'''simple docstring'''





					return self._value




			@staticmethod
			def 			snake_case__					(       a_							:			RandomizedHeapNode[T] | None	,	a_							:			RandomizedHeapNode[T] | None			):
					'''simple docstring'''





					if not roota:
							return roota

					if not roota:
							return roota

					if roota.value > roota.value:
							__UpperCAmelCase							, __UpperCAmelCase   :					int													=     roota, roota

					if random.choice([True, False]			):
							__UpperCAmelCase							, __UpperCAmelCase   :					Union[str, Any]													=     roota.right, roota.left

					__UpperCAmelCase   :					Optional[Any]													=     RandomizedHeapNode.merge(roota.left	,	a_			)

					return roota

class 		UpperCAmelCase__      (      Generic[T]       ):




			'''simple docstring'''






			def __init__(       self							:			Tuple	,	a_							:			Iterable[T] | None = ()			):
					'''simple docstring'''





					__UpperCAmelCase   :					RandomizedHeapNode[T] | None													=     None

					if data:
							for item in data:
									self.insert(a_			)




			def 			snake_case__					(       self							:			List[str]	,	a_							:			T			):
					'''simple docstring'''





					__UpperCAmelCase   :					Optional[Any]													=     RandomizedHeapNode.merge(self._root	,	RandomizedHeapNode(a_			)			)




			def 			snake_case__					(       self							:			Optional[int]			):
					'''simple docstring'''





					__UpperCAmelCase   :					Optional[Any]													=     self.top()

					if self._root is None:
							return None

					__UpperCAmelCase   :					Tuple													=     RandomizedHeapNode.merge(self._root.left	,	self._root.right			)

					return result




			def 			snake_case__					(       self							:			Any			):
					'''simple docstring'''





					if not self._root:
							raise IndexError('''Can\'t get top element for the empty heap.'''			)
					return self._root.value




			def 			snake_case__					(       self							:			Optional[Any]			):
					'''simple docstring'''





					__UpperCAmelCase   :					str													=     None




			def 			snake_case__					(       self							:			Tuple			):
					'''simple docstring'''





					__UpperCAmelCase   :					List[Any]													=     []
					while self:
							result.append(self.pop()			)

					return result




			def __bool__(       self							:			List[Any]			):
					'''simple docstring'''





					return self._root is not None


if __name__ == "__main__":
				import doctest

				doctest.testmod()





