
'''simple docstring'''
import sys
from random import randint






class 				UpperCAmelCase_			:





 def __init__(      self	,    lowercase_	,    lowercase_):
     snake_case_	:      Tuple            =  key
     snake_case_	:      Any            =  freq





 def __str__(      self):
     return F'Node(key={self.key}, freq={self.freq})'


def   UpperCamelCase_       (		__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE    ):
    """simple docstring"""






    if i > j or i < 0 or j > len(__SCREAMING_SNAKE_CASE    ) - 1:
        return

    snake_case_	:      Optional[Any]            =  root[i][j]
    if parent == -1:  # root does not have a parent
        print(f'{key[node]} is the root of the binary search tree.'    )
    elif is_left:
        print(f'{key[node]} is the left child of key {parent}.'    )
    else:
        print(f'{key[node]} is the right child of key {parent}.'    )

    print_binary_search_tree(__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						node - 1,						key[node],						__SCREAMING_SNAKE_CASE    )
    print_binary_search_tree(__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						node + 1,						__SCREAMING_SNAKE_CASE,						key[node],						__SCREAMING_SNAKE_CASE    )

def   UpperCamelCase_       (		__SCREAMING_SNAKE_CASE    ):
    """simple docstring"""






    nodes.sort(key=lambda __SCREAMING_SNAKE_CASE    : node.key    )

    snake_case_	:      int            =  len(__SCREAMING_SNAKE_CASE    )

    snake_case_	:      Optional[Any]            =  [nodes[i].key for i in range(__SCREAMING_SNAKE_CASE    )]
    snake_case_	:      Union[str, Any]            =  [nodes[i].freq for i in range(__SCREAMING_SNAKE_CASE    )]

    # This 2D array stores the overall tree cost (which's as minimized as possible);
    # for a single key, cost is equal to frequency of the key.
    snake_case_	:      Optional[Any]            =  [[freqs[i] if i == j else 0 for j in range(__SCREAMING_SNAKE_CASE    )] for i in range(__SCREAMING_SNAKE_CASE    )]
    # sum[i][j] stores the sum of key frequencies between i and j inclusive in nodes
    # array
    snake_case_	:      Union[str, Any]            =  [[freqs[i] if i == j else 0 for j in range(__SCREAMING_SNAKE_CASE    )] for i in range(__SCREAMING_SNAKE_CASE    )]
    # stores tree roots that will be used later for constructing binary search tree
    snake_case_	:      Tuple            =  [[i if i == j else 0 for j in range(__SCREAMING_SNAKE_CASE    )] for i in range(__SCREAMING_SNAKE_CASE    )]

    for interval_length in range(2,						n + 1    ):
        for i in range(n - interval_length + 1    ):
            snake_case_	:      Optional[int]            =  i + interval_length - 1

            snake_case_	:      int            =  sys.maxsize  # set the value to "infinity"
            snake_case_	:      Union[str, Any]            =  total[i][j - 1] + freqs[j]

            # Apply Knuth's optimization
            # Loop without optimization: for r in range(i, j + 1):
            for r in range(root[i][j - 1],						root[i + 1][j] + 1    ):  # r is a temporal root
                snake_case_	:      Union[str, Any]            =  dp[i][r - 1] if r != i else 0  # optimal cost for left subtree
                snake_case_	:      int            =  dp[r + 1][j] if r != j else 0  # optimal cost for right subtree
                snake_case_	:      Optional[Any]            =  left + total[i][j] + right

                if dp[i][j] > cost:
                    snake_case_	:      List[Any]            =  cost
                    snake_case_	:      int            =  r

    print("Binary search tree nodes:"    )
    for node in nodes:
        print(__SCREAMING_SNAKE_CASE    )

    print(f'\nThe cost of optimal BST for given tree nodes is {dp[0][n - 1]}.'    )
    print_binary_search_tree(__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						0,						n - 1,						-1,						__SCREAMING_SNAKE_CASE    )

def   UpperCamelCase_       (		):
    """simple docstring"""






    snake_case_	:      Optional[Any]            =  [Node(__SCREAMING_SNAKE_CASE,						randint(1,						5_0    )    ) for i in range(1_0,						0,						-1    )]
    find_optimal_binary_search_tree(__SCREAMING_SNAKE_CASE    )


if __name__ == "__main__":
  main()






'''simple docstring'''
import math

def   UpperCamelCase_       (		__SCREAMING_SNAKE_CASE    ):
    """simple docstring"""






    snake_case_	:      Tuple            =  math.log(2,						1_0    ) * number
    snake_case_	:      Dict            =  round((log_number - int(__SCREAMING_SNAKE_CASE    )),						1_5    )

    return difference

def   UpperCamelCase_       (		__SCREAMING_SNAKE_CASE = 6_7_8_9_1_0    ):
    """simple docstring"""






    snake_case_	:      Any            =  9_0
    snake_case_	:      Optional[int]            =  0

    snake_case_	:      Union[str, Any]            =  math.log(1.23,						1_0    )
    snake_case_	:      Dict            =  math.log(1.24,						1_0    )
    snake_case_	:      List[Any]            =  0

    while position < number:
        snake_case_	:      Union[str, Any]            =  log_difference(__SCREAMING_SNAKE_CASE    )

        if difference >= upper_limit:
            power_iterator += 9_3

        elif difference < lower_limit:
            power_iterator += 1_9_6

        else:
            snake_case_	:      Optional[Any]            =  power_iterator
            power_iterator += 1_9_6
            position += 1

    return previous_power


if __name__ == "__main__":
  import doctest

  doctest.testmod()

  print(f"""{solution() = }""")




