code: 
  
  
  
  
  
  '''simple docstring'''
  
  
  
  
  
  
  
  from typing import Any
  
  import numpy as np
  
  
  
  
  
  class 					_UpperCAmelCase							:
  		def __init__(						self,__SCREAMING_SNAKE_CASE,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					if np.any(tableau[:, -1],where=tableau[:, -1] < 0    ):
  								raise ValueError("""RHS must be > 0"""    )
  
  					__lowerCAmelCase      		=       tableau
  					__lowerCAmelCase		, __lowerCAmelCase      		=       tableau.shape
  
  					# Number of decision variables x1, x2, x3...
  					__lowerCAmelCase      		=       n_vars
  
  					# Number of artificial variables to be minimised
  					__lowerCAmelCase      		=       len(np.where(tableau[self.n_vars : -1] == -1    )[0]    )
  
  					# 2 if there are >= or == constraints (nonstandard), 1 otherwise (std)
  					__lowerCAmelCase      		=       (self.n_art_vars > 0) + 1
  
  					# Number of slack variables added to make inequalities into equalities
  					__lowerCAmelCase      		=       self.n_rows - self.n_stages
  
  					# Objectives for each stage
  					__lowerCAmelCase      		=       ["""max"""]
  
  					# In two stage simplex, first minimise then maximise
  					if self.n_art_vars:
  								self.objectives.append("""min"""    )
  
  					__lowerCAmelCase      		=       [""""""]
  
  					# Index of current pivot row and column
  					__lowerCAmelCase      		=       None
  					__lowerCAmelCase      		=       None
  
  					# Does objective row only contain (non)-negative values?
  					__lowerCAmelCase      		=       False
  		@staticmethod
  		def 						lowerCamelCase__    (						*__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''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
  					__lowerCAmelCase      		=       ["""x""", """s""", """a"""]
  					__lowerCAmelCase      		=       []
  					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 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       self.objectives[-1]
  
  					# Find entries of highest magnitude in objective rows
  					__lowerCAmelCase      		=       (objective == """min""") - (objective == """max""")
  					__lowerCAmelCase      		=       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:
  								__lowerCAmelCase      		=       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
  					__lowerCAmelCase      		=       slice(self.n_stages,self.n_rows    )
  
  					# RHS
  					__lowerCAmelCase      		=       tableau[s, -1]
  
  					# Elements of pivot column within slice
  					__lowerCAmelCase      		=       tableau[s, col_idx]
  
  					# Array filled with nans
  					__lowerCAmelCase      		=       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
  					__lowerCAmelCase      		=       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
  					__lowerCAmelCase      		=       np.nanargmin(__SCREAMING_SNAKE_CASE    ) + self.n_stages
  					return row_idx, col_idx
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE,__SCREAMING_SNAKE_CASE,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       tableau[row_idx].copy()
  
  					__lowerCAmelCase      		=       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
  					__lowerCAmelCase      		=       piv_row
  					return tableau
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					self.objectives.pop()
  
  					if not self.objectives:
  								return tableau
  
  					# Slice containing ids for artificial columns
  					__lowerCAmelCase      		=       slice(-self.n_art_vars - 1,-1    )
  
  					# Delete the artificial variable columns
  					__lowerCAmelCase      		=       np.delete(__SCREAMING_SNAKE_CASE,__SCREAMING_SNAKE_CASE,axis=1    )
  
  					# Delete the objective row of the first stage
  					__lowerCAmelCase      		=       np.delete(__SCREAMING_SNAKE_CASE,0,axis=0    )
  
  					__lowerCAmelCase      		=       1
  					self.n_rows -= 1
  					__lowerCAmelCase      		=       0
  					__lowerCAmelCase      		=       False
  					return tableau
  		def 						lowerCamelCase__    (						self    ):
  
  
  
  					'''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:
  											__lowerCAmelCase      		=       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    )
  
  								__lowerCAmelCase		, __lowerCAmelCase      		=       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
  											__lowerCAmelCase      		=       self.change_stage(self.tableau    )
  								else:
  											__lowerCAmelCase      		=       self.pivot(self.tableau,__SCREAMING_SNAKE_CASE,__SCREAMING_SNAKE_CASE    )
  					return {}
  
  
  
  
  
  
  
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       {"""P""": abs(tableau[0, -1]    )}
  
  					for i in range(self.n_vars    ):
  								# Gives ids of nonzero entries in the ith column
  								__lowerCAmelCase      		=       np.nonzero(tableau[:, i]    )
  								__lowerCAmelCase      		=       len(nonzero[0]    )
  
  								# First entry in the nonzero ids
  								__lowerCAmelCase      		=       nonzero[0][0]
  								__lowerCAmelCase      		=       tableau[nonzero_rowidx, i]
  
  								# If there is only one nonzero value in column, which is one
  								if n_nonzero == nonzero_val == 1:
  											__lowerCAmelCase      		=       tableau[nonzero_rowidx, -1]
  											__lowerCAmelCase      		=       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()
  
  
style_context: 
  
  
  
  
  
  '''simple docstring'''
  
  
  
  
  
  
  
  import math
  import sys
  
  
  
  
  
  class 					_UpperCAmelCase							:
  		def __init__(						self    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       0
  					__lowerCAmelCase      		=       []
  					__lowerCAmelCase      		=       {}  # To store the pos of node in array
  		def 						lowerCamelCase__    (						self    ):
  
  
  
  					'''simple docstring'''
  
  					return self.cur_size == 0
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       self.left(__SCREAMING_SNAKE_CASE    )
  					__lowerCAmelCase      		=       self.right(__SCREAMING_SNAKE_CASE    )
  					if lc < self.cur_size and self.array(__SCREAMING_SNAKE_CASE    )[0] < self.array(__SCREAMING_SNAKE_CASE    )[0]:
  								__lowerCAmelCase      		=       lc
  					else:
  								__lowerCAmelCase      		=       idx
  					if rc < self.cur_size and self.array(__SCREAMING_SNAKE_CASE    )[0] < self.array(__SCREAMING_SNAKE_CASE    )[0]:
  								__lowerCAmelCase      		=       rc
  					if smallest != idx:
  								self.swap(__SCREAMING_SNAKE_CASE,__SCREAMING_SNAKE_CASE    )
  								self.min_heapify(__SCREAMING_SNAKE_CASE    )
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       self.cur_size
  					self.cur_size += 1
  					self.array.append((sys.maxsize, tup[1])    )
  					self.decrease_key((sys.maxsize, tup[1]),tup[0]    )
  		def 						lowerCamelCase__    (						self    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       self.array[0][1]
  					__lowerCAmelCase      		=       self.array[self.cur_size - 1]
  					self.cur_size -= 1
  					self.min_heapify(1    )
  					del self.pos[min_node]
  					return min_node
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					return 2 * i + 1
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					return 2 * i + 2
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					return math.floor(i / 2    )
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       j
  					__lowerCAmelCase      		=       i
  					__lowerCAmelCase      		=       self.array[i]
  					__lowerCAmelCase      		=       self.array[j]
  					__lowerCAmelCase      		=       temp
  
  
  
  
  
  
  
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       self.pos[tup[1]]
  					# assuming the new_d is atmost old_d
  					__lowerCAmelCase      		=       (new_d, tup[1])
  					while idx > 0 and self.array[self.par(__SCREAMING_SNAKE_CASE    )][0] > self.array[idx][0]:
  								self.swap(__SCREAMING_SNAKE_CASE,self.par(__SCREAMING_SNAKE_CASE    )    )
  								__lowerCAmelCase      		=       self.par(__SCREAMING_SNAKE_CASE    )
  
  
  
  
  
  class 					_UpperCAmelCase							:
  		def __init__(						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       {}  # To store graph: u -> (v,w)
  					__lowerCAmelCase      		=       num  # Number of nodes in graph
  					# To store the distance from source vertex
  					__lowerCAmelCase      		=       [0] * self.num_nodes
  					__lowerCAmelCase      		=       [-1] * self.num_nodes  # To store the path
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE,__SCREAMING_SNAKE_CASE,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					if u in self.adjList:
  								self.adjList[u].append((v, w)    )
  					else:
  								__lowerCAmelCase      		=       [(v, w)]
  
  					# Assuming undirected graph
  					if v in self.adjList:
  								self.adjList[v].append((u, w)    )
  					else:
  								__lowerCAmelCase      		=       [(u, w)]
  		def 						lowerCamelCase__    (						self    ):
  
  
  
  					'''simple docstring'''
  
  					for u in self.adjList:
  								print(__SCREAMING_SNAKE_CASE,"""->""",""" -> """.join(str(f'{v}({w})'    ) for v, w in self.adjList[u]    )    )
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       [-1] * self.num_nodes
  					# src is the source node
  					__lowerCAmelCase      		=       0
  					__lowerCAmelCase      		=       PriorityQueue()
  					q.insert((0, src)    )  # (dist from src, node)
  					for u in self.adjList:
  								if u != src:
  											__lowerCAmelCase      		=       sys.maxsize  # Infinity
  											__lowerCAmelCase      		=       -1
  
  					while not q.is_empty():
  								__lowerCAmelCase      		=       q.extract_min()  # Returns node with the min dist from source
  								# Update the distance of all the neighbours of u and
  								# if their prev dist was INFINITY then push them in Q
  								for v, w in self.adjList[u]:
  											__lowerCAmelCase      		=       self.dist[u] + w
  											if self.dist[v] > new_dist:
  														if self.dist[v] == sys.maxsize:
  																	q.insert((new_dist, v)    )
  														else:
  																	q.decrease_key((self.dist[v], v),__SCREAMING_SNAKE_CASE    )
  														__lowerCAmelCase      		=       new_dist
  														__lowerCAmelCase      		=       u
  
          # Show the shortest distances from src
  					self.show_distances(__SCREAMING_SNAKE_CASE    )
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					print(f'Distance from node: {src}'    )
  					for u in range(self.num_nodes    ):
  								print(f'Node {u} has distance: {self.dist[u]}'    )
  
  
  
  
  
  
  
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       []
  					__lowerCAmelCase      		=       0
  					__lowerCAmelCase      		=       dest
  					# Backtracking from dest to src
  					while self.par[temp] != -1:
  								path.append(__SCREAMING_SNAKE_CASE    )
  								if temp != src:
  											for v, w in self.adjList[temp]:
  														if v == self.par[temp]:
  																	cost += w
  																	break
  								__lowerCAmelCase      		=       self.par[temp]
  					path.append(__SCREAMING_SNAKE_CASE    )
  					path.reverse()
  
  					print(f'----Path to reach {dest} from {src}----'    )
  					for u in path:
  								print(f'{u}',end=""" """    )
  								if u != dest:
  											print("""-> """,end=""""""    )
  
  					print("""\nTotal cost of path: """,__SCREAMING_SNAKE_CASE    )
  
  
  if __name__ == "__main__":
  	_a      :				str											=					Graph(9)
  	graph.add_edge(0, 1, 4)
  	graph.add_edge(0, 7, 8)
  	graph.add_edge(1, 2, 8)
  	graph.add_edge(1, 7, 1_1)
  	graph.add_edge(2, 3, 7)
  	graph.add_edge(2, 8, 2)
  	graph.add_edge(2, 5, 4)
  	graph.add_edge(3, 4, 9)
  	graph.add_edge(3, 5, 1_4)
  	graph.add_edge(4, 5, 1_0)
  	graph.add_edge(5, 6, 2)
  	graph.add_edge(6, 7, 1)
  	graph.add_edge(6, 8, 6)
  	graph.add_edge(7, 8, 7)
  	graph.show_graph()
  	graph.dijkstra(0)
  	graph.show_path(0, 4)
  
  # OUTPUT
  # 0 -> 1(4) -> 7(8)
  # 1 -> 0(4) -> 2(8) -> 7(11)
  # 7 -> 0(8) -> 1(11) -> 6(1) -> 8(7)
  # 2 -> 1(8) -> 3(7) -> 8(2) -> 5(4)
  # 3 -> 2(7) -> 4(9) -> 5(14)
  # 8 -> 2(2) -> 6(6) -> 7(7)
  # 5 -> 2(4) -> 3(14) -> 4(10) -> 6(2)
  # 4 -> 3(9) -> 5(10)
  # 6 -> 5(2) -> 7(1) -> 8(6)
  # Distance from node: 0
  # Node 0 has distance: 0
  # Node 1 has distance: 4
  # Node 2 has distance: 12
  # Node 3 has distance: 19
  # Node 4 has distance: 21
  # Node 5 has distance: 11
  # Node 6 has distance: 9
  # Node 7 has distance: 8
  # Node 8 has distance: 14
  # ----Path to reach 4 from 0----
  # 0 -> 7 -> 6 -> 5 -> 4
  # Total cost of path:  21
  
  