



"""simple docstring"""
def       _snake_case						(						UpperCamelCase						:		Tuple			,		UpperCamelCase						:		Optional[int]						):
 UpperCAmelCase	:					Tuple	=	0
 UpperCAmelCase	:					str	=	len(UpperCamelCase						) - 1

 while left <= right:
  # avoid divided by 0 during interpolation
  if sorted_collection[left] == sorted_collection[right]:
   if sorted_collection[left] == item:
    return left
   else:
    return None

  UpperCAmelCase	:					List[Any]	=	left + ((item - sorted_collection[left]) * (right - left)) // (
      sorted_collection[right] - sorted_collection[left]
  )

  # out of range check
  if point < 0 or point >= len(UpperCamelCase						):
   return None

  UpperCAmelCase	:					Tuple	=	sorted_collection[point]
  if current_item == item:
   return point
  else:
   if point < left:
    UpperCAmelCase	:					List[str]	=	left
    UpperCAmelCase	:					Union[str, Any]	=	point
   elif point > right:
    UpperCAmelCase	:					Union[str, Any]	=	right
    UpperCAmelCase	:					List[str]	=	point
   else:
    if item < current_item:
     UpperCAmelCase	:					Tuple	=	point - 1
    else:
     UpperCAmelCase	:					Any	=	point + 1
 return None

def       _snake_case						(						UpperCamelCase						:		int			,		UpperCamelCase						:		Tuple			,		UpperCamelCase						:		int			,		UpperCamelCase						:		Dict						):

 # avoid divided by 0 during interpolation
 if sorted_collection[left] == sorted_collection[right]:
  if sorted_collection[left] == item:
   return left
  else:
   return None

 UpperCAmelCase	:					Optional[int]	=	left + ((item - sorted_collection[left]) * (right - left)) // (
     sorted_collection[right] - sorted_collection[left]
 )

 # out of range check
 if point < 0 or point >= len(UpperCamelCase						):
  return None

 if sorted_collection[point] == item:
  return point
 elif point < left:
  return interpolation_search_by_recursion(UpperCamelCase			,		UpperCamelCase			,		UpperCamelCase			,		UpperCamelCase						)
 elif point > right:
  return interpolation_search_by_recursion(UpperCamelCase			,		UpperCamelCase			,		UpperCamelCase			,		UpperCamelCase						)
 else:
  if sorted_collection[point] > item:
   return interpolation_search_by_recursion(
       UpperCamelCase			,		UpperCamelCase			,		UpperCamelCase			,		point - 1						)
  else:
   return interpolation_search_by_recursion(
       UpperCamelCase			,		UpperCamelCase			,		point + 1			,		UpperCamelCase						)

def       _snake_case						(						UpperCamelCase						:		Optional[Any]						):
 if collection != sorted(UpperCamelCase						):
  raise ValueError("""Collection must be ascending sorted"""						)
 return True


if __name__ == "__main__":
      import sys
      A:   Any								=		0
      if debug == 1:
            A:   Optional[int]								=		[1_0, 3_0, 4_0, 4_5, 5_0, 6_6, 7_7, 9_3]
            try:
                  __assert_sorted(collection)
            except ValueError:
                  sys.exit("Sequence must be ascending sorted to apply interpolation search")
            A:   str								=		6_7

      A:   List[Any]								=		interpolation_search(collection, target)
      if result is not None:
            print(f"""{target} found at positions: {result}""")
      else:
            print("Not found")




"""simple docstring"""







from __future__ import annotations

from collections import namedtuple
from dataclasses import dataclass


@dataclass
class       SCREAMING_SNAKE_CASE__				:
       """simple docstring"""


       a  :				int
       a  :				TreeNode | None   =None
       a  :				TreeNode | None   =None

lowerCAmelCase__      				=   namedtuple('''CoinsDistribResult''', '''moves excess''')
def 					a__			(				SCREAMING_SNAKE_CASE    :    TreeNode | None				):


  '''simple docstring'''







  if root is None:
    return 0

  # Validation
  def count_nodes(SCREAMING_SNAKE_CASE    :    TreeNode | None				) -> int:
    if node is None:
      return 0

    return count_nodes(node.left				) + count_nodes(node.right				) + 1

  def count_coins(SCREAMING_SNAKE_CASE    :    TreeNode | None				) -> int:
    if node is None:
      return 0

    return count_coins(node.left				) + count_coins(node.right				) + node.data

  if count_nodes(SCREAMING_SNAKE_CASE				) != count_coins(SCREAMING_SNAKE_CASE				):
    raise ValueError("The nodes number should be same as the number of coins"				)

  # Main calculation
  def get_distrib(SCREAMING_SNAKE_CASE    :    TreeNode | None				) -> CoinsDistribResult:

    if node is None:
      return CoinsDistribResult(0				,      1				)

    lowerCAmelCase						,  lowerCAmelCase     :     List[str]								=							get_distrib(node.left				)
    lowerCAmelCase						,  lowerCAmelCase     :     List[str]								=							get_distrib(node.right				)

    lowerCAmelCase     :     Union[str, Any]								=							1 - left_distrib_excess
    lowerCAmelCase     :     List[Any]								=							1 - right_distrib_excess

    lowerCAmelCase     :     int								=							(
        left_distrib_moves
        + right_distrib_moves
        + abs(SCREAMING_SNAKE_CASE				)
        + abs(SCREAMING_SNAKE_CASE				)
    )
    lowerCAmelCase     :     int								=							node.data - coins_to_left - coins_to_right

    return CoinsDistribResult(SCREAMING_SNAKE_CASE				,      SCREAMING_SNAKE_CASE				)

  return get_distrib(SCREAMING_SNAKE_CASE				)[0]


if __name__ == "__main__":
     import doctest

     doctest.testmod()





