



from typing import Any

import numpy as np




class 				lowerCAmelCase__	:







					def __init__(							self   :  List[str]      ,		_A   :  np.ndarray      ,		_A   :  int):
											# Check if RHS is negative
											if np.any(tableau[:, -1]      ,		where=tableau[:, -1] < 0):
																	raise ValueError("RHS must be > 0")

											A__						:				Dict         =			tableau
											A__     ,							A__						:				Optional[int]         =			tableau.shape

											# Number of decision variables x1, x2, x3...
											A__						:				Optional[int]         =			n_vars

											# Number of artificial variables to be minimised
											A__						:				Union[str, Any]         =			len(np.where(tableau[self.n_vars : -1] == -1)[0])

											# 2 if there are >= or == constraints (nonstandard), 1 otherwise (std)
											A__						:				Union[str, Any]         =			(self.n_art_vars > 0) + 1

											# Number of slack variables added to make inequalities into equalities
											A__						:				List[Any]         =			self.n_rows - self.n_stages

											# Objectives for each stage
											A__						:				List[str]         =			["max"]

											# In two stage simplex, first minimise then maximise
											if self.n_art_vars:
																	self.objectives.append("min")

											A__						:				Optional[int]         =			[""]

											# Index of current pivot row and column
											A__						:				List[Any]         =			None
											A__						:				Any         =			None

											# Does objective row only contain (non)-negative values?
											A__						:				str         =			False







					@staticmethod
					def 			_lowercase      (							*_A   :  int):
											if len(_A) != 3:
																	raise ValueError("Must provide n_vars, n_slack, and n_art_vars")

											if not all(x >= 0 and isinstance(_A      ,		_A) for x in args):
																	raise ValueError("All arguments must be non-negative integers")

											# decision | slack | artificial
											A__						:				List[Any]         =			["x", "s", "a"]
											A__						:				Optional[Any]         =			[]
											for i in range(3):
																	for j in range(args[i]):
																							titles.append(string_starts[i] + str(j + 1))
											titles.append("RHS")
											return titles







					def 			_lowercase      (							self   :  Optional[Any]      ,		_A   :  np.ndarray):
											A__						:				Union[str, Any]         =			self.objectives[-1]

											# Find entries of highest magnitude in objective rows
											A__						:				int         =			(objective == "min") - (objective == "max")
											A__						:				str         =			np.argmax(sign * tableau[0, : self.n_vars])

											# Choice is only valid if below 0 for maximise, and above for minimise
											if sign * self.tableau[0, col_idx] <= 0:
																	A__						:				Any         =			True
																	return 0, 0

											# Pivot row is chosen as having the lowest quotient when elements of
											# the pivot column divide the right-hand side

											# Slice excluding the objective rows
											A__						:				List[str]         =			slice(self.n_stages      ,		self.n_rows)

											# RHS
											A__						:				Union[str, Any]         =			tableau[s, -1]

											# Elements of pivot column within slice
											A__						:				Tuple         =			tableau[s, col_idx]

											# Array filled with nans
											A__						:				Optional[Any]         =			np.full(self.n_rows - self.n_stages      ,		np.nan)

											# If element in pivot column is greater than zeron_stages, return
											# quotient or nan otherwise
											A__						:				Optional[Any]         =			np.divide(_A      ,		_A      ,		out=_A      ,		where=divisor > 0)

											# Arg of minimum quotient excluding the nan values. n_stages is added
											# to compensate for earlier exclusion of objective columns
											A__						:				List[str]         =			np.nanargmin(_A) + self.n_stages
											return row_idx, col_idx







					def 			_lowercase      (							self   :  Union[str, Any]      ,		_A   :  np.ndarray      ,		_A   :  int      ,		_A   :  int):
											A__						:				List[Any]         =			tableau[row_idx].copy()

											A__						:				Optional[int]         =			piv_row[col_idx]

											# Entry becomes 1
											piv_row *= 1 / piv_val

											# Variable in pivot column becomes basic, ie the only non-zero entry
											for idx, coeff in enumerate(tableau[:, col_idx]):
																	tableau[idx] += -coeff * piv_row
											A__						:				Optional[int]         =			piv_row
											return tableau







					def 			_lowercase      (							self   :  List[Any]      ,		_A   :  np.ndarray):
											self.objectives.pop()

											if not self.objectives:
																	return tableau

											# Slice containing ids for artificial columns
											A__						:				Union[str, Any]         =			slice(-self.n_art_vars - 1      ,		-1)

											# Delete the artificial variable columns
											A__						:				Any         =			np.delete(_A      ,		_A      ,		axis=1)

											# Delete the objective row of the first stage
											A__						:				str         =			np.delete(_A      ,		0      ,		axis=0)

											A__						:				List[Any]         =			1
											self.n_rows -= 1
											A__						:				List[Any]         =			0
											A__						:				Optional[Any]         =			False
											return tableau







					def 			_lowercase      (							self   :  Dict):
											# Stop simplex algorithm from cycling.
											for _ in range(100):
																	# Completion of each stage removes an objective. If both stages
																	# are complete, then no objectives are left
																	if not self.objectives:
																							A__						:				Tuple         =			self.generate_col_titles(
																							    self.n_vars      ,		self.n_slack      ,		self.n_art_vars)

																							# Find the values of each variable at optimal solution
																							return self.interpret_tableau(self.tableau      ,		self.col_titles)

																	A__     ,							A__						:				Dict         =			self.find_pivot(self.tableau)

																	# If there are no more negative values in objective row
																	if self.stop_iter:
																							# Delete artificial variable columns and rows. Update attributes
																							A__						:				Tuple         =			self.change_stage(self.tableau)
																	else:
																							A__						:				Optional[int]         =			self.pivot(self.tableau      ,		_A      ,		_A)
											return {}







					def 			_lowercase      (							self   :  Any      ,		_A   :  np.ndarray      ,		_A   :  list[str]):
											A__						:				Optional[Any]         =			{"P": abs(tableau[0, -1])}

											for i in range(self.n_vars):
																	# Gives ids of nonzero entries in the ith column
																	A__						:				Union[str, Any]         =			np.nonzero(tableau[:, i])
																	A__						:				List[str]         =			len(nonzero[0])

																	# First entry in the nonzero ids
																	A__						:				int         =			nonzero[0][0]
																	A__						:				Dict         =			tableau[nonzero_rowidx, i]

																	# If there is only one nonzero value in column, which is one
																	if n_nonzero == nonzero_val == 1:
																							A__						:				int         =			tableau[nonzero_rowidx, -1]
																							A__						:				Dict         =			rhs_val

        # Check for basic variables
											for title in col_titles:
																	# Don't add RHS or slack variables to output dict
																	if title[0] not in "R-s-a":
																							output_dict.setdefault(_A      ,		0)
											return output_dict


if __name__ == "__main__":
		import doctest

		doctest.testmod()




class 				lowerCAmelCase__	:







					def __init__(							self   :  Union[str, Any]):
											A__						:				Optional[int]         =			[0]
											A__						:				Any         =			0







					def 			_lowercase      (							self   :  Optional[int]      ,		_A   :  int):
											A__						:				Optional[Any]         =			self.__heap[i]
											while i // 2 > 0:
																	if self.__heap[i] > self.__heap[i // 2]:
																							A__						:				Union[str, Any]         =			self.__heap[i // 2]
																							A__						:				List[str]         =			temporary
																	i //= 2







					def 			_lowercase      (							self   :  Union[str, Any]      ,		_A   :  int):
											self.__heap.append(_A)
											self.__size += 1
											self.__swap_up(self.__size)







					def 			_lowercase      (							self   :  Any      ,		_A   :  int):
											while self.__size >= 2 * i:
																	if 2 * i + 1 > self.__size:
																							A__						:				str         =			2 * i
																	else:
																							if self.__heap[2 * i] > self.__heap[2 * i + 1]:
																													A__						:				Tuple         =			2 * i
																							else:
																													A__						:				Tuple         =			2 * i + 1
																	A__						:				Any         =			self.__heap[i]
																	if self.__heap[i] < self.__heap[bigger_child]:
																							A__						:				List[str]         =			self.__heap[bigger_child]
																							A__						:				Tuple         =			temporary
																	A__						:				Tuple         =			bigger_child







					def 			_lowercase      (							self   :  int):
											A__						:				str         =			self.__heap[1]
											A__						:				Tuple         =			self.__heap[self.__size]
											self.__size -= 1
											self.__heap.pop()
											self.__swap_down(1)
											return max_value







					@property
					def 			_lowercase      (							self   :  Any):
											return self.__heap[1:]







					def __len__(							self   :  Union[str, Any]):
											return self.__size


if __name__ == "__main__":
		import doctest

		doctest.testmod()
		# create an instance of BinaryHeap
		snake_case     :    List[Any]						=    BinaryHeap()
		binary_heap.insert(6)
		binary_heap.insert(1_0)
		binary_heap.insert(1_5)
		binary_heap.insert(1_2)
		# pop root(max-values because it is max heap)
		print(binary_heap.pop())  # 15
		print(binary_heap.pop())  # 12
		# get the list and size after operations
		print(binary_heap.get_list)
		print(len(binary_heap))