




'''simple docstring'''







from __future__ import annotations

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

_lowerCamelCase				:		str				      =    TypeVar('T', bound=bool)





class 	lowerCamelCase_				(			Generic[T]				):




					'''simple docstring'''






					def __init__(       self,    __UpperCamelCase					):
							_snake_case      			=			value
							_snake_case      			=			None
							_snake_case      			=			None






					@property
					def 							__SCREAMING_SNAKE_CASE      (       self					):
							return self._value






					@staticmethod
					def 							__SCREAMING_SNAKE_CASE      (       __UpperCamelCase,    __UpperCamelCase					):
							if not roota:
									return roota

							if not roota:
									return roota

							if roota.value > roota.value:
									_snake_case			,       _snake_case      			=			roota, roota

							if random.choice([True, False]					):
									_snake_case			,       _snake_case      			=			roota.right, roota.left

							_snake_case      			=			RandomizedHeapNode.merge(roota.left,    _lowerCAmelCase					)

							return roota





class 	lowerCamelCase_				(			Generic[T]				):




					'''simple docstring'''






					def __init__(       self,    __UpperCamelCase = ()					):
							_snake_case      			=			None

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






					def 							__SCREAMING_SNAKE_CASE      (       self,    __UpperCamelCase					):
							_snake_case      			=			RandomizedHeapNode.merge(self._root,    RandomizedHeapNode(_lowerCAmelCase					)					)






					def 							__SCREAMING_SNAKE_CASE      (       self					):
							_snake_case      			=			self.top()

							if self._root is None:
									return None

							_snake_case      			=			RandomizedHeapNode.merge(self._root.left,    self._root.right					)

							return result






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






					def 							__SCREAMING_SNAKE_CASE      (       self					):
							_snake_case      			=			None






					def 							__SCREAMING_SNAKE_CASE      (       self					):
							_snake_case      			=			[]
							while self:
									result.append(self.pop()					)

							return result






					def __bool__(       self					):
							return self._root is not None


if __name__ == "__main__":
						import doctest

						doctest.testmod()










import numpy as np
def 				UpperCAmelCase__		(					_SCREAMING_SNAKE_CASE :						Tuple					)->Union[str, Any]:
      _lowerCAmelCase      , _lowerCAmelCase			      =	a.shape
      _lowerCAmelCase			      =	min(_SCREAMING_SNAKE_CASE  ,					_SCREAMING_SNAKE_CASE					)
      _lowerCAmelCase			      =	np.eye(_SCREAMING_SNAKE_CASE					)
      _lowerCAmelCase			      =	a.copy()

      for k in range(t - 1					):
            # select a column of modified matrix A':
            _lowerCAmelCase			      =	r[k:, [k]]
            # construct first basis vector
            _lowerCAmelCase			      =	np.zeros_like(_SCREAMING_SNAKE_CASE					)
            _lowerCAmelCase			      =	1.0
            # determine scaling factor
            _lowerCAmelCase			      =	np.linalg.norm(_SCREAMING_SNAKE_CASE					)
            # construct vector v for Householder reflection
            _lowerCAmelCase			      =	x + np.sign(x[0]					) * alpha * ea
            v /= np.linalg.norm(_SCREAMING_SNAKE_CASE					)

            # construct the Householder matrix
            _lowerCAmelCase			      =	np.eye(m - k					) - 2.0 * v @ v.T
            # pad with ones and zeros as necessary
            _lowerCAmelCase			      =	np.block([[np.eye(_SCREAMING_SNAKE_CASE					), np.zeros((k, m - k)					)], [np.zeros((m - k, k)					), q_k]]					)

            _lowerCAmelCase			      =	q @ q_k.T
            _lowerCAmelCase			      =	q_k @ r

      return q, r


if __name__ == "__main__":
    import doctest

    doctest.testmod()