from typing import Any

import numpy as np

class 						_SCREAMING_SNAKE_CASE				:






		"""simple docstring"""







		def __init__(      self     :					Optional[Any]	,       __SCREAMING_SNAKE_CASE     :					np.ndarray	,       __SCREAMING_SNAKE_CASE     :					int ):






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

					_snake_case														=   tableau
					_snake_case							,			_snake_case														=   tableau.shape

					# Number of decision variables x1, x2, x3...
					_snake_case														=   n_vars

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

					# 2 if there are >= or == constraints (nonstandard), 1 otherwise (std)
					_snake_case														=   (self.n_art_vars > 0) + 1

					# Number of slack variables added to make inequalities into equalities
					_snake_case														=   self.n_rows - self.n_stages

					# Objectives for each stage
					_snake_case														=   ["max"]

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

					_snake_case														=   [""]

					# Index of current pivot row and column
					_snake_case														=   None
					_snake_case														=   None

					# Does objective row only contain (non)-negative values?
					_snake_case														=   False



		@staticmethod
		def 					_a					(      *__SCREAMING_SNAKE_CASE     :					int ):






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

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

					# decision | slack | artificial
					_snake_case														=   ["x", "s", "a"]
					_snake_case														=   []
					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 					_a					(      self     :					Any	,       __SCREAMING_SNAKE_CASE     :					np.ndarray ):






					"""simple docstring"""
					_snake_case														=   self.objectives[-1]

					# Find entries of highest magnitude in objective rows
					_snake_case														=   (objective == "min") - (objective == "max")
					_snake_case														=   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														=   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														=   slice(self.n_stages	,       self.n_rows )

					# RHS
					_snake_case														=   tableau[s, -1]

					# Elements of pivot column within slice
					_snake_case														=   tableau[s, col_idx]

					# Array filled with nans
					_snake_case														=   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														=   np.divide(__SCREAMING_SNAKE_CASE	,       __SCREAMING_SNAKE_CASE	,       out=__SCREAMING_SNAKE_CASE	,       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														=   np.nanargmin(__SCREAMING_SNAKE_CASE ) + self.n_stages
					return row_idx, col_idx



		def 					_a					(      self     :					List[str]	,       __SCREAMING_SNAKE_CASE     :					np.ndarray	,       __SCREAMING_SNAKE_CASE     :					int	,       __SCREAMING_SNAKE_CASE     :					int ):






					"""simple docstring"""
					_snake_case														=   tableau[row_idx].copy()

					_snake_case														=   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														=   piv_row
					return tableau



		def 					_a					(      self     :					Tuple	,       __SCREAMING_SNAKE_CASE     :					np.ndarray ):






					"""simple docstring"""
					self.objectives.pop()

					if not self.objectives:
								return tableau

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

					# Delete the artificial variable columns
					_snake_case														=   np.delete(__SCREAMING_SNAKE_CASE	,       __SCREAMING_SNAKE_CASE	,       axis=1 )

					# Delete the objective row of the first stage
					_snake_case														=   np.delete(__SCREAMING_SNAKE_CASE	,       0	,       axis=0 )

					_snake_case														=   1
					self.n_rows -= 1
					_snake_case														=   0
					_snake_case														=   False
					return tableau



		def 					_a					(      self     :					Optional[Any] ):






					"""simple docstring"""
					# 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:
											_snake_case														=   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														=   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														=   self.change_stage(self.tableau )
								else:
											_snake_case														=   self.pivot(self.tableau	,       __SCREAMING_SNAKE_CASE	,       __SCREAMING_SNAKE_CASE )
					return {}







		def 					_a					(      self     :					Optional[Any]	,       __SCREAMING_SNAKE_CASE     :					np.ndarray	,       __SCREAMING_SNAKE_CASE     :					list[str] ):






					"""simple docstring"""
					_snake_case														=   {"P": abs(tableau[0, -1] )}

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

								# First entry in the nonzero ids
								_snake_case														=   nonzero[0][0]
								_snake_case														=   tableau[nonzero_rowidx, i]

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


if __name__ == "__main__":
			import doctest

			doctest.testmod()


def  UpperCAmelCase__(    lowerCamelCase_	):



			"""simple docstring"""





			_snake_case														=   len(lowerCamelCase_	)
			_snake_case														=   [0] * length
			_snake_case														=   [[False for i in range(lowerCamelCase_	)] for j in range(lowerCamelCase_	)]
			for i, c in enumerate(lowerCamelCase_	):
						_snake_case														=   i
						for j in range(i + 1	):
									if c == string[j] and (i - j < 2 or is_palindromic[j + 1][i - 1]):
												_snake_case														=   True
												_snake_case														=   min(lowerCamelCase_				,						0 if j == 0 else (cut[j - 1] + 1)	)
						_snake_case														=   mincut
			return cut[length - 1]


if __name__ == "__main__":
			_SCREAMING_SNAKE_CASE:    Dict     						=							input('''Enter the string: ''').strip()
			_SCREAMING_SNAKE_CASE:    str     						=							find_minimum_partitions(s)
			print(F"Minimum number of partitions required for the '{s}' is {ans}")

