




'''simple docstring'''





import numpy as np


def 				UpperCamelCase	(  snake_case   ):




   '''simple docstring'''

   SCREAMING_SNAKE_CASE__						,				SCREAMING_SNAKE_CASE__			:				Optional[int]       						= a.shape
   SCREAMING_SNAKE_CASE__			:				Tuple       						= min(snake_case	,     snake_case   )
   SCREAMING_SNAKE_CASE__			:				Tuple       						= np.eye(snake_case   )
   SCREAMING_SNAKE_CASE__			:				str       						= a.copy()

   for k in range(t - 1   ):
      # select a column of modified matrix A':
      SCREAMING_SNAKE_CASE__			:				str       						= r[k:, [k]]
      # construct first basis vector
      SCREAMING_SNAKE_CASE__			:				Tuple       						= np.zeros_like(snake_case   )
      SCREAMING_SNAKE_CASE__			:				Any       						= 1.0
      # determine scaling factor
      SCREAMING_SNAKE_CASE__			:				str       						= np.linalg.norm(snake_case   )
      # construct vector v for Householder reflection
      SCREAMING_SNAKE_CASE__			:				List[Any]       						= x + np.sign(x[0]   ) * alpha * ea
      v /= np.linalg.norm(snake_case   )

      # construct the Householder matrix
      SCREAMING_SNAKE_CASE__			:				Optional[int]       						= np.eye(m - k   ) - 2.0 * v @ v.T
      # pad with ones and zeros as necessary
      SCREAMING_SNAKE_CASE__			:				Optional[int]       						= np.block([[np.eye(snake_case   ), np.zeros((k, m - k)   )], [np.zeros((m - k, k)   ), q_k]]   )

      SCREAMING_SNAKE_CASE__			:				Any       						= q @ q_k.T
      SCREAMING_SNAKE_CASE__			:				Union[str, Any]       						= q_k @ r

   return q, r


if __name__ == "__main__":
       import doctest

       doctest.testmod()







'''simple docstring'''





from __future__ import annotations

import sys






class 						lowercase     :






      def __init__(self ,  a__ ,  a__					)       ->   List[str]:
         SCREAMING_SNAKE_CASE__			:				str       						= letter
         SCREAMING_SNAKE_CASE__			:				int       						= freq
         SCREAMING_SNAKE_CASE__			:				dict[str, str]       						= {}






      def __repr__(self					)       ->   str:
         return f"""{self.letter}:{self.freq}"""






class 						lowercase     :






      def __init__(self ,  a__ ,  a__ ,  a__					)       ->   Union[str, Any]:
         SCREAMING_SNAKE_CASE__			:				int       						= freq
         SCREAMING_SNAKE_CASE__			:				Letter | TreeNode       						= left
         SCREAMING_SNAKE_CASE__			:				Letter | TreeNode       						= right


def 				UpperCamelCase	(  snake_case   ):




   '''simple docstring'''

   SCREAMING_SNAKE_CASE__			:				dict[str, int]       						= {}
   with open(snake_case   ) as f:
      while True:
         SCREAMING_SNAKE_CASE__			:				List[str]       						= f.read(1   )
         if not c:
            break
         SCREAMING_SNAKE_CASE__			:				Optional[Any]       						= chars[c] + 1 if c in chars else 1
   return sorted((Letter(snake_case	,     snake_case   ) for c, f in chars.items())	,     key=lambda snake_case   : x.freq   )


def 				UpperCamelCase	(  snake_case   ):




   '''simple docstring'''

   SCREAMING_SNAKE_CASE__			:				list[Letter | TreeNode]       						= letters  # type: ignore
   while len(snake_case   ) > 1:
      SCREAMING_SNAKE_CASE__			:				Optional[Any]       						= response.pop(0   )
      SCREAMING_SNAKE_CASE__			:				Union[str, Any]       						= response.pop(0   )
      SCREAMING_SNAKE_CASE__			:				Dict       						= left.freq + right.freq
      SCREAMING_SNAKE_CASE__			:				Any       						= TreeNode(snake_case	,     snake_case	,     snake_case   )
      response.append(snake_case   )
      response.sort(key=lambda snake_case   : x.freq   )
   return response[0]


def 				UpperCamelCase	(  snake_case	,     snake_case   ):




   '''simple docstring'''

   if isinstance(snake_case	,     snake_case   ):
      SCREAMING_SNAKE_CASE__			:				Optional[int]       						= bitstring
      return [root]
   SCREAMING_SNAKE_CASE__			:				TreeNode       						= root  # type: ignore
   SCREAMING_SNAKE_CASE__			:				int       						= []
   letters += traverse_tree(treenode.left	,     bitstring + '''0'''   )
   letters += traverse_tree(treenode.right	,     bitstring + '''1'''   )
   return letters


def 				UpperCamelCase	(  snake_case   ):




   '''simple docstring'''

   SCREAMING_SNAKE_CASE__			:				Optional[int]       						= parse_file(snake_case   )
   SCREAMING_SNAKE_CASE__			:				List[Any]       						= build_tree(snake_case   )
   SCREAMING_SNAKE_CASE__			:				List[str]       						= {
       k: v for letter in traverse_tree(snake_case	,     ''''''   ) for k, v in letter.bitstring.items()
   }
   print(f"""Huffman Coding  of {file_path}: """   )
   with open(snake_case   ) as f:
      while True:
         SCREAMING_SNAKE_CASE__			:				Tuple       						= f.read(1   )
         if not c:
            break
         print(letters[c]	,     end=''' '''   )
   print()


if __name__ == "__main__":
       # pass the file path to the huffman function
       huffman(sys.argv[1])

