

'''simple docstring'''







class 						A__ :



    def __init__(   self						,			UpperCamelCase__ = ""						,			UpperCamelCase__ = False    )  -> int:





           '''simple docstring'''
           # Mapping from the first character of the prefix of the node
           A_		   = {}

           # A node will be a leaf if the tree contains its word
           A_		   = is_leaf

           A_		   = prefix



    def    snake_case_     (   self						,			UpperCamelCase__    )  -> Tuple:





           '''simple docstring'''
           A_		   = 0
           for q, w in zip(self.prefix						,			_a    ):
                  if q != w:
                         break

                  x += 1

           return self.prefix[:x], self.prefix[x:], word[x:]



    def    snake_case_     (   self						,			UpperCamelCase__    )  -> Union[str, Any]:





           '''simple docstring'''
           for word in words:
                  self.insert(_a    )



    def    snake_case_     (   self						,			UpperCamelCase__    )  -> str:





           '''simple docstring'''
           # Case 1: If the word is the prefix of the node
           # Solution: We set the current node as leaf
           if self.prefix == word:
                  A_		   = True

           # Case 2: The node has no edges that have a prefix to the word
           # Solution: We create an edge from the current node to a new one
           # containing the word
           elif word[0] not in self.nodes:
                  A_		   = RadixNode(prefix=_a						,			is_leaf=_a    )

           else:
                  A_		   = self.nodes[word[0]]
                  A_		   = incoming_node.match(
                      _a    )

                  # Case 3: The node prefix is equal to the matching
                  # Solution: We insert remaining word on the next node
                  if remaining_prefix == "":
                         self.nodes[matching_string[0]].insert(_a    )

                  # Case 4: The word is greater equal to the matching
                  # Solution: Create a node in between both nodes, change
                  # prefixes and add the new node for the remaining word
                  else:
                         A_		   = remaining_prefix

                         A_		   = self.nodes[matching_string[0]]
                         A_		   = RadixNode(_a						,			_a    )
                         A_		   = aux_node

                         if remaining_word == "":
                                A_		   = True
                         else:
                                self.nodes[matching_string[0]].insert(_a    )



    def    snake_case_     (   self						,			UpperCamelCase__    )  -> Optional[Any]:





           '''simple docstring'''
           A_		   = self.nodes.get(word[0]						,			_a    )
           if not incoming_node:
                  return False
           else:
                  A_		   = incoming_node.match(
                      _a    )
                  # If there is remaining prefix, the word can't be on the tree
                  if remaining_prefix != "":
                         return False
                  # This applies when the word and the prefix are equal
                  elif remaining_word == "":
                         return incoming_node.is_leaf
                  # We have word remaining so we check the next node
                  else:
                         return incoming_node.find(_a    )



    def    snake_case_     (   self						,			UpperCamelCase__    )  -> List[Any]:





           '''simple docstring'''
           A_		   = self.nodes.get(word[0]						,			_a    )
           if not incoming_node:
                  return False
           else:
                  A_		   = incoming_node.match(
                      _a    )
                  # If there is remaining prefix, the word can't be on the tree
                  if remaining_prefix != "":
                         return False
                  # We have word remaining so we check the next node
                  elif remaining_word != "":
                         return incoming_node.delete(_a    )
                  else:
                         # If it is not a leaf, we don't have to delete
                         if not incoming_node.is_leaf:
                                return False
                         else:
                                # We delete the nodes if no edges go from it
                                if len(incoming_node.nodes    ) == 0:
                                       del self.nodes[word[0]]
                                       # We merge the current node with its only child
                                       if len(self.nodes    ) == 1 and not self.is_leaf:
                                              A_		   = list(self.nodes.values()    )[0]
                                              A_		   = merging_node.is_leaf
                                              self.prefix += merging_node.prefix
                                              A_		   = merging_node.nodes
                    # If there is more than 1 edge, we just mark it as non-leaf
                                elif len(incoming_node.nodes    ) > 1:
                                       A_		   = False
                                # If there is 1 edge, we merge it with its child
                                else:
                                       A_		   = list(incoming_node.nodes.values()    )[0]
                                       A_		   = merging_node.is_leaf
                                       incoming_node.prefix += merging_node.prefix
                                       A_		   = merging_node.nodes

                                return True







    def    snake_case_     (   self						,			UpperCamelCase__ = 0    )  -> int:





           '''simple docstring'''
           if self.prefix != "":
                  print("""-""" * height						,			self.prefix						,			"""  (leaf)""" if self.is_leaf else """"""    )

           for value in self.nodes.values():
                  value.print_tree(height + 1    )

def  UpperCAmelCase__      ( )     ->     bool:
       A_		   = "banana bananas bandana band apple all beast".split()
       A_		   = RadixNode()
       root.insert_many(_snake_case			)

       assert all(root.find(_snake_case			) for word in words			)
       assert not root.find("""bandanas"""			)
       assert not root.find("""apps"""			)
       root.delete("""all"""			)
       assert not root.find("""all"""			)
       root.delete("""banana"""			)
       assert not root.find("""banana"""			)
       assert root.find("""bananas"""			)

       return True
def  UpperCAmelCase__      ( )     ->     None:
       assert test_trie()
def  UpperCAmelCase__      ( )     ->     None:
       A_		   = RadixNode()
       A_		   = "banana bananas bandanas bandana band apple all beast".split()
       root.insert_many(_snake_case			)

       print("""Words:""",    _snake_case			)
       print("""Tree:"""			)
       root.print_tree()


if __name__ == "__main__":
      main()



'''simple docstring'''







from __future__ import annotations

import numpy as np
from numpy import floataa
from numpy.typing import NDArray
def  UpperCAmelCase__      ( UpperCAmelCase__,    UpperCAmelCase__,    UpperCAmelCase__,    UpperCAmelCase__,    )     ->     list[float]:
       A_   ,     A_		   = coefficient_matrix.shape
       A_   ,     A_		   = constant_matrix.shape

       if rowsa != colsa:
              A_		   = F'''Coefficient matrix dimensions must be nxn but received {rowsa}x{colsa}'''
              raise ValueError(UpperCAmelCase__			)

       if colsa != 1:
              A_		   = F'''Constant matrix must be nx1 but received {rowsa}x{colsa}'''
              raise ValueError(UpperCAmelCase__			)

       if rowsa != rowsa:
              A_		   = (
                  """Coefficient and constant matrices dimensions must be nxn and nx1 but """
                  F'''received {rowsa}x{colsa} and {rowsa}x{colsa}'''
              )
              raise ValueError(UpperCAmelCase__			)

       if len(UpperCAmelCase__			) != rowsa:
              A_		   = (
                  """Number of initial values must be equal to number of rows in coefficient """
                  F'''matrix but received {len(UpperCAmelCase__			)} and {rowsa}'''
              )
              raise ValueError(UpperCAmelCase__			)

       if iterations <= 0:
              raise ValueError("""Iterations must be at least 1"""			)

       A_		   = np.concatenate(
           (coefficient_matrix, constant_matrix),    axis=1			)

       A_   ,     A_		   = table.shape

       strictly_diagonally_dominant(UpperCAmelCase__			)

       # Iterates the whole matrix for given number of times
       for _ in range(UpperCAmelCase__			):
              A_		   = []
              for row in range(UpperCAmelCase__			):
                     A_		   = 0
                     for col in range(UpperCAmelCase__			):
                            if col == row:
                                   A_		   = table[row][col]
                            elif col == cols - 1:
                                   A_		   = table[row][col]
                            else:
                                   temp += (-1) * table[row][col] * init_val[col]
                     A_		   = (temp + val) / denom
                     new_val.append(UpperCAmelCase__			)
              A_		   = new_val

       return [float(UpperCAmelCase__			) for i in new_val]
def  UpperCAmelCase__      ( UpperCAmelCase__			)     ->     bool:
       A_   ,     A_		   = table.shape

       A_		   = True

       for i in range(0,    UpperCAmelCase__			):
              A_		   = 0
              for j in range(0,    cols - 1			):
                     if i == j:
                            continue
                     else:
                            total += table[i][j]

              if table[i][i] <= total:
                     raise ValueError("""Coefficient matrix is not strictly diagonally dominant"""			)

       return is_diagonally_dominant


# Test Cases
if __name__ == "__main__":
      import doctest

      doctest.testmod()
