






'''simple docstring'''
from __future__ import annotations

from random import random






class      __SCREAMING_SNAKE_CASE      :




 def __init__(      self	:							Union[str, Any]					,      __lowercase	:							int | None = None      )	->						int:
       SCREAMING_SNAKE_CASE__     :				Any									=value
       SCREAMING_SNAKE_CASE__     :				str									=random()
       SCREAMING_SNAKE_CASE__     :				Node | None									=None
       SCREAMING_SNAKE_CASE__     :				Node | None									=None




 def __repr__(      self	:							List[Any]      )	->						str:
       from pprint import pformat

       if self.left is None and self.right is None:
             return F"'{self.value}: {self.prior:.5}'"
       else:
             return pformat(
                 {F"{self.value}: {self.prior:.5}": (self.left, self.right)}					,      indent=1      )




 def __str__(      self	:							Tuple      )	->						str:
       SCREAMING_SNAKE_CASE__     :				int									=str(self.value      ) + ''' '''
       SCREAMING_SNAKE_CASE__     :				str									=str(self.left or ''''''      )
       SCREAMING_SNAKE_CASE__     :				Tuple									=str(self.right or ''''''      )
       return value + left + right





def  _a(  UpperCamelCase__				:						Node | None,	UpperCamelCase__				:						int							):




      '''simple docstring'''






      if root is None:  # None tree is split into 2 Nones
            return None, None
      elif root.value is None:
            return None, None
      else:
            if value < root.value:
                  SCREAMING_SNAKE_CASE__		,	SCREAMING_SNAKE_CASE__     :				str									=split(root.left,	UpperCamelCase__							)
                  return left, root
            else:
                  SCREAMING_SNAKE_CASE__		,	SCREAMING_SNAKE_CASE__     :				List[Any]									=split(root.right,	UpperCamelCase__							)
                  return root, right





def  _a(  UpperCamelCase__				:						Node | None,	UpperCamelCase__				:						Node | None							):




      '''simple docstring'''






      if (not left) or (not right):  # If one node is None, return the other
            return left or right
      elif left.prior < right.prior:
            SCREAMING_SNAKE_CASE__     :				List[Any]									=merge(left.right,	UpperCamelCase__							)
            return left
      else:
            SCREAMING_SNAKE_CASE__     :				int									=merge(UpperCamelCase__,	right.left							)
            return right





def  _a(  UpperCamelCase__				:						Node | None,	UpperCamelCase__				:						int							):




      '''simple docstring'''






      SCREAMING_SNAKE_CASE__     :				Dict									=Node(UpperCamelCase__							)
      SCREAMING_SNAKE_CASE__		,	SCREAMING_SNAKE_CASE__     :				Optional[int]									=split(UpperCamelCase__,	UpperCamelCase__							)
      return merge(merge(UpperCamelCase__,	UpperCamelCase__							),	UpperCamelCase__							)





def  _a(  UpperCamelCase__				:						Node | None,	UpperCamelCase__				:						int							):




      '''simple docstring'''






      SCREAMING_SNAKE_CASE__		,	SCREAMING_SNAKE_CASE__     :				Tuple									=split(UpperCamelCase__,	value - 1							)
      SCREAMING_SNAKE_CASE__		,	SCREAMING_SNAKE_CASE__     :				Union[str, Any]									=split(UpperCamelCase__,	UpperCamelCase__							)
      return merge(UpperCamelCase__,	UpperCamelCase__							)





def  _a(  UpperCamelCase__				:						Node | None							):




      '''simple docstring'''






      if not root:  # None
            return
      else:
            inorder(root.left							)
            print(root.value,	end=''','''							)
            inorder(root.right							)





def  _a(  UpperCamelCase__				:						Node | None,	UpperCamelCase__				:						str							):




      '''simple docstring'''






      for arg in args.split():
            if arg[0] == "+":
                  SCREAMING_SNAKE_CASE__     :				List[str]									=insert(UpperCamelCase__,	int(arg[1:]							)							)

            elif arg[0] == "-":
                  SCREAMING_SNAKE_CASE__     :				List[Any]									=erase(UpperCamelCase__,	int(arg[1:]							)							)

            else:
                  print('''Unknown command'''							)

      return root





def  _a(  ):




      '''simple docstring'''






      SCREAMING_SNAKE_CASE__     :				Union[str, Any]									=None
      print(
          '''enter numbers to create a tree, + value to add value into treap, '''
          '''- value to erase all nodes with value. \'q\' to quit. '''							)

      SCREAMING_SNAKE_CASE__     :				List[str]									=input()
      while args != "q":
            SCREAMING_SNAKE_CASE__     :				Any									=interact_treap(UpperCamelCase__,	UpperCamelCase__							)
            print(UpperCamelCase__							)
            SCREAMING_SNAKE_CASE__     :				Dict									=input()

      print('''good by!'''							)


if __name__ == "__main__":
  import doctest

  doctest.testmod()
  main()







'''simple docstring'''
from __future__ import annotations

import numpy as np
from numpy import floataa
from numpy.typing import NDArray





def  _a(  UpperCamelCase__				:						NDArray[floataa],	UpperCamelCase__				:						NDArray[floataa],	UpperCamelCase__				:						list[int],	UpperCamelCase__				:						int,	):




      '''simple docstring'''






      SCREAMING_SNAKE_CASE__		,	SCREAMING_SNAKE_CASE__     :				Tuple									=coefficient_matrix.shape
      SCREAMING_SNAKE_CASE__		,	SCREAMING_SNAKE_CASE__     :				Tuple									=constant_matrix.shape

      if rowsa != colsa:
            SCREAMING_SNAKE_CASE__     :				Any									=f"Coefficient matrix dimensions must be nxn but received {rowsa}x{colsa}"
            raise ValueError(UpperCamelCase__							)

      if colsa != 1:
            SCREAMING_SNAKE_CASE__     :				str									=f"Constant matrix must be nx1 but received {rowsa}x{colsa}"
            raise ValueError(UpperCamelCase__							)

      if rowsa != rowsa:
            SCREAMING_SNAKE_CASE__     :				str									=(
                '''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:
            SCREAMING_SNAKE_CASE__     :				Union[str, Any]									=(
                '''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'''							)

      SCREAMING_SNAKE_CASE__     :				NDArray[floataa]									=np.concatenate(
          (coefficient_matrix, constant_matrix),	axis=1							)

      SCREAMING_SNAKE_CASE__		,	SCREAMING_SNAKE_CASE__     :				Any									=table.shape

      strictly_diagonally_dominant(UpperCamelCase__							)

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

      return [float(UpperCamelCase__							) for i in new_val]





def  _a(  UpperCamelCase__				:						NDArray[floataa]							):




      '''simple docstring'''






      SCREAMING_SNAKE_CASE__		,	SCREAMING_SNAKE_CASE__     :				Optional[int]									=table.shape

      SCREAMING_SNAKE_CASE__     :				Any									=True

      for i in range(0,	UpperCamelCase__							):
            SCREAMING_SNAKE_CASE__     :				int									=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()