





'''simple docstring'''






from typing import Any

import numpy as np




class    _A   :

		def __init__(      self				:      Optional[Any]     ,			__magic_name__				:      np.ndarray     ,			__magic_name__				:      int )				->						None:


					"""simple docstring"""

					if np.any(tableau[:, -1]     ,			where=tableau[:, -1] < 0 ):
								raise ValueError("""RHS must be > 0""" )

					__snake_case				:			List[str]       			=    tableau
					__snake_case			,    __snake_case				:			Optional[Any]       			=    tableau.shape

					# Number of decision variables x1, x2, x3...
					__snake_case				:			Union[str, Any]       			=    n_vars

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

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

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

					# Objectives for each stage
					__snake_case				:			Any       			=    ["""max"""]

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

					__snake_case				:			List[Any]       			=    [""""""]

					# Index of current pivot row and column
					__snake_case				:			List[Any]       			=    None
					__snake_case				:			Union[str, Any]       			=    None

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

		@staticmethod
		def   lowercase__    (      *__magic_name__				:      int )				->						list[str]:


					"""simple docstring"""

					if len(__magic_name__ ) != 3:
								raise ValueError("""Must provide n_vars, n_slack, and n_art_vars""" )

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

					# decision | slack | artificial
					__snake_case				:			Union[str, Any]       			=    ["""x""", """s""", """a"""]
					__snake_case				:			Optional[int]       			=    []
					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[int]     ,			__magic_name__				:      np.ndarray )				->						tuple[Any, Any]:


					"""simple docstring"""

					__snake_case				:			List[str]       			=    self.objectives[-1]

					# Find entries of highest magnitude in objective rows
					__snake_case				:			List[str]       			=    (objective == """min""") - (objective == """max""")
					__snake_case				:			Union[str, Any]       			=    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:
								__snake_case				:			Optional[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
					__snake_case				:			str       			=    slice(self.n_stages     ,			self.n_rows )

					# RHS
					__snake_case				:			Optional[Any]       			=    tableau[s, -1]

					# Elements of pivot column within slice
					__snake_case				:			Dict       			=    tableau[s, col_idx]

					# Array filled with nans
					__snake_case				:			Union[str, 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
					__snake_case				:			Union[str, Any]       			=    np.divide(__magic_name__     ,			__magic_name__     ,			out=__magic_name__     ,			where=divisor > 0 )

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

		def   lowercase__    (      self				:      List[str]     ,			__magic_name__				:      np.ndarray     ,			__magic_name__				:      int     ,			__magic_name__				:      int )				->						np.ndarray:


					"""simple docstring"""

					__snake_case				:			List[Any]       			=    tableau[row_idx].copy()

					__snake_case				:			Dict       			=    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
					__snake_case				:			Union[str, Any]       			=    piv_row
					return tableau

		def   lowercase__    (      self				:      Optional[Any]     ,			__magic_name__				:      np.ndarray )				->						np.ndarray:


					"""simple docstring"""

					self.objectives.pop()

					if not self.objectives:
								return tableau

					# Slice containing ids for artificial columns
					__snake_case				:			Dict       			=    slice(-self.n_art_vars - 1     ,			-1 )

					# Delete the artificial variable columns
					__snake_case				:			Optional[int]       			=    np.delete(__magic_name__     ,			__magic_name__     ,			axis=1 )

					# Delete the objective row of the first stage
					__snake_case				:			int       			=    np.delete(__magic_name__     ,			0     ,			axis=0 )

					__snake_case				:			int       			=    1
					self.n_rows -= 1
					__snake_case				:			List[str]       			=    0
					__snake_case				:			Any       			=    False
					return tableau

		def   lowercase__    (      self				:      Union[str, Any] )				->						dict[Any, Any]:


					"""simple docstring"""

					for _ in range(1_00 ):
								# Completion of each stage removes an objective. If both stages
								# are complete, then no objectives are left
								if not self.objectives:
											__snake_case				:			int       			=    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 )

								__snake_case			,    __snake_case				:			List[Any]       			=    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
											__snake_case				:			List[Any]       			=    self.change_stage(self.tableau )
								else:
											__snake_case				:			List[Any]       			=    self.pivot(self.tableau     ,			__magic_name__     ,			__magic_name__ )
					return {}

		def   lowercase__    (      self				:      Union[str, Any]     ,			__magic_name__				:      np.ndarray     ,			__magic_name__				:      list[str] )				->						dict[str, float]:


					"""simple docstring"""

					__snake_case				:			Optional[int]       			=    {"""P""": abs(tableau[0, -1] )}

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

								# First entry in the nonzero ids
								__snake_case				:			Any       			=    nonzero[0][0]
								__snake_case				:			List[str]       			=    tableau[nonzero_rowidx, i]

								# If there is only one nonzero value in column, which is one
								if n_nonzero == nonzero_val == 1:
											__snake_case				:			Optional[Any]       			=    tableau[nonzero_rowidx, -1]
											__snake_case				:			str       			=    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(__magic_name__     ,			0 )
					return output_dict


if __name__ == "__main__":
	import doctest

	doctest.testmod()









'''simple docstring'''






def 				_a	(					_lowerCamelCase   )   ->							bool:



			"""simple docstring"""


			__snake_case				:			Optional[Any]       			=    [list(_lowerCamelCase   ) for row in matrix]
			if matrix and isinstance(_lowerCamelCase						,  _lowerCamelCase   ):
						if isinstance(matrix[0]						,  _lowerCamelCase   ):
									__snake_case				:			str       			=    0
									for row in matrix:
												if prev_len == 0:
															__snake_case				:			Dict       			=    len(_lowerCamelCase   )
															__snake_case				:			List[Any]       			=    True
												else:
															__snake_case				:			Union[str, Any]       			=    prev_len == len(_lowerCamelCase   )
						else:
									__snake_case				:			List[Any]       			=    True
			else:
						__snake_case				:			List[str]       			=    False

			return result







def 				_a	(					_lowerCamelCase   )   ->							None:



			"""simple docstring"""


			if check_matrix(_lowerCamelCase   ) and len(_lowerCamelCase   ) > 0:
						__snake_case				:			List[str]       			=    [list(_lowerCamelCase   ) for row in a]
						__snake_case				:			Optional[Any]       			=    len(_lowerCamelCase   )
						if isinstance(a[0]						,  _lowerCamelCase   ):
									__snake_case				:			Tuple       			=    len(a[0]   )
						else:
									for dat in a:
												print(_lowerCamelCase   )
									return

						# horizotal printing increasing
						for i in range(0						,  _lowerCamelCase   ):
									print(a[0][i]   )
						# vertical printing down
						for i in range(1						,  _lowerCamelCase   ):
									print(a[i][mat_col - 1]   )
						# horizotal printing decreasing
						if mat_row > 1:
									for i in range(mat_col - 2						,  -1						,  -1   ):
												print(a[mat_row - 1][i]   )
        # vertical printing up
						for i in range(mat_row - 2						,  0						,  -1   ):
									print(a[i][0]   )
						__snake_case				:			Union[str, Any]       			=    [row[1 : mat_col - 1] for row in a[1 : mat_row - 1]]
						if len(_lowerCamelCase   ) > 0:
									spiral_print_clockwise(_lowerCamelCase   )
						else:
									return
			else:
						print("""Not a valid matrix"""   )
						return







def 				_a	(					_lowerCamelCase   )   ->							list[int]:



			"""simple docstring"""


			if matrix:
						return list(matrix.pop(0   )   ) + spiral_traversal(list(zip(*_lowerCamelCase   )   )[::-1]   )
			else:
						return []


# driver code
if __name__ == "__main__":
	import doctest

	doctest.testmod()

	__UpperCamelCase										=  [[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]]
	spiral_print_clockwise(a)


