




from __future__ import annotations

__lowerCAmelCase		:    Optional[Any]	      =		[2, 1, 4, 5, 6, 0, 8, 9, 1, 2, 0, 6, 4, 2, 0, 6, 5, 3, 2, 7]




class    snake_case__							:
    """simple docstring"""



    def __init__(				self    :					Optional[Any] ,				__lowerCamelCase    :					int							) ->					None:
       a   =      -1
       a   =      -1
       a   =      [-1] * length
       a   =      None
       a   =      None





    def __repr__(				self    :					List[Any]							) ->					str:
       return f"""Node(min_value={self.minn} max_value={self.maxx})"""




def       __magic_name__    (			A		: list[int]       ):






   '''simple docstring'''







   a   =      Node(len(A       )       )
   a   ,		a   =      min(A       ), max(A       )
   # Leaf node case where the node contains only one unique value
   if root.minn == root.maxx:
      return root
   a   =      (root.minn + root.maxx) // 2

   a   =      []
   a   =      []

   for index, num in enumerate(A       ):
      if num <= pivot:
         left_arr.append(A       )
      else:
         right_arr.append(A       )
      a   =      len(A       )
   a   =      build_tree(A       )
   a   =      build_tree(A       )
   return root




def       __magic_name__    (			A		: Node | None,  A		: int,  A		: int       ):






   '''simple docstring'''







   if index < 0 or node is None:
      return 0
   # Leaf node cases
   if node.minn == node.maxx:
      return index + 1 if node.minn == num else 0
   a   =      (node.minn + node.maxx) // 2
   if num <= pivot:
      # go the left subtree and map index to the left subtree
      return rank_till_index(node.left,  A,  node.map_left[index] - 1       )
   else:
      # go to the right subtree and map index to the right subtree
      return rank_till_index(node.right,  A,  index - node.map_left[index]       )




def       __magic_name__    (			A		: Node | None,  A		: int,  A		: int,  A		: int       ):






   '''simple docstring'''







   if start > end:
      return 0
   a   =      rank_till_index(A,  A,  A       )
   a   =      rank_till_index(A,  A,  start - 1       )
   return rank_till_end - rank_before_start




def       __magic_name__    (			A		: Node | None,  A		: int,  A		: int,  A		: int       ):






   '''simple docstring'''







   if index > (end - start) or start > end or node is None:
      return -1
   # Leaf node case
   if node.minn == node.maxx:
      return node.minn
   # Number of elements in the left subtree in interval [start, end]
   a   =      node.map_left[end] - (
       node.map_left[start - 1] if start else 0
   )
   if num_elements_in_left_tree > index:
      return quantile(
          node.left,  A,  (node.map_left[start - 1] if start else 0),  node.map_left[end] - 1,  )
   else:
      return quantile(
          node.right,  index - num_elements_in_left_tree,  start - (node.map_left[start - 1] if start else 0),  end - node.map_left[end],  )




def       __magic_name__    (			A		: Node | None,  A		: int,  A		: int,  A		: int,  A		: int       ):






   '''simple docstring'''







   if (
       start > end
       or node is None
       or start_num > end_num
       or node.minn > end_num
       or node.maxx < start_num
   ):
      return 0
   if start_num <= node.minn and node.maxx <= end_num:
      return end - start + 1
   a   =      range_counting(
       node.left,  (node.map_left[start - 1] if start else 0),  node.map_left[end] - 1,  A,  A,  )
   a   =      range_counting(
       node.right,  start - (node.map_left[start - 1] if start else 0),  end - node.map_left[end],  A,  A,  )
   return left + right


if __name__ == "__main__":
  import doctest

  doctest.testmod()












from typing import Any

import numpy as np




class    snake_case__							:
				"""simple docstring"""



				def __init__(				self    :					Tuple ,				__lowerCamelCase    :					np.ndarray ,				__lowerCamelCase    :					int							) ->					None:
							# Check if RHS is negative
							if np.any(tableau[:, -1] ,				where=tableau[:, -1] < 0							):
										raise ValueError("RHS must be > 0"							)

							a   =      tableau
							a   ,		a   =      tableau.shape

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

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

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

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

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

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

							a   =      [""]

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

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





				@staticmethod
				def 							__UpperCAmelCase							(				*__lowerCamelCase    :					int							) ->					list[str]:
							if len(__lowerCamelCase							) != 3:
										raise ValueError("Must provide n_vars, n_slack, and n_art_vars"							)

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

							# decision | slack | artificial
							a   =      ["x", "s", "a"]
							a   =      []
							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 							__UpperCAmelCase							(				self    :					Tuple ,				__lowerCamelCase    :					np.ndarray							) ->					tuple[Any, Any]:
							a   =      self.objectives[-1]

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

							# RHS
							a   =      tableau[s, -1]

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

							# Array filled with nans
							a   =      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   =      np.divide(__lowerCamelCase ,				__lowerCamelCase ,				out=__lowerCamelCase ,				where=divisor > 0							)

							# Arg of minimum quotient excluding the nan values. n_stages is added
							# to compensate for earlier exclusion of objective columns
							a   =      np.nanargmin(__lowerCamelCase							) + self.n_stages
							return row_idx, col_idx





				def 							__UpperCAmelCase							(				self    :					List[Any] ,				__lowerCamelCase    :					np.ndarray ,				__lowerCamelCase    :					int ,				__lowerCamelCase    :					int							) ->					np.ndarray:
							a   =      tableau[row_idx].copy()

							a   =      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   =      piv_row
							return tableau





				def 							__UpperCAmelCase							(				self    :					Tuple ,				__lowerCamelCase    :					np.ndarray							) ->					np.ndarray:
							self.objectives.pop()

							if not self.objectives:
										return tableau

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

							# Delete the artificial variable columns
							a   =      np.delete(__lowerCamelCase ,				__lowerCamelCase ,				axis=1							)

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

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





				def 							__UpperCAmelCase							(				self    :					Any							) ->					dict[Any, Any]:
							# Stop simplex algorithm from cycling.
							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:
													a   =      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   =      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   =      self.change_stage(self.tableau							)
										else:
													a   =      self.pivot(self.tableau ,				__lowerCamelCase ,				__lowerCamelCase							)
							return {}





				def 							__UpperCAmelCase							(				self    :					int ,				__lowerCamelCase    :					np.ndarray ,				__lowerCamelCase    :					list[str]							) ->					dict[str, float]:
							a   =      {"P": abs(tableau[0, -1]							)}

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

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

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


if __name__ == "__main__":
		import doctest

		doctest.testmod()






