



def  _snake_case			(							__snake_case       ,	__snake_case     )  ->  Optional[Any]:






      '''simple docstring'''







      UpperCAmelCase_	:						str					   =		[]
      UpperCAmelCase_	:						Dict					   =		len(__snake_case     )
      for i in range(__snake_case     ):
            UpperCAmelCase_	:						int					   =		[]
            for j in range(__snake_case     ):
                  UpperCAmelCase_	:						int					   =		0
                  for k in range(__snake_case     ):
                        UpperCAmelCase_	:						Optional[int]					   =		val + matrix_a[i][k] * matrix_b[k][j]
                  list_a.append(__snake_case     )
            matrix_c.append(__snake_case     )
      return matrix_c







def  _snake_case			(							__snake_case     )  ->  str:






      '''simple docstring'''







      return [[int(row == column     ) for column in range(__snake_case     )] for row in range(__snake_case     )]







def  _snake_case			(							__snake_case     )  ->  Dict:






      '''simple docstring'''







      if n <= 1:
            return n
      UpperCAmelCase_	:						int					   =		identity(2     )
      UpperCAmelCase_	:						Tuple					   =		[[1, 1], [1, 0]]
      UpperCAmelCase_	:						int					   =		n - 1
      while n > 0:
            if n % 2 == 1:
                  UpperCAmelCase_	:						Dict					   =		multiply(__snake_case       ,	__snake_case     )
            UpperCAmelCase_	:						Tuple					   =		multiply(__snake_case       ,	__snake_case     )
            UpperCAmelCase_	:						Optional[Any]					   =		int(n / 2     )
      return res_matrix[0][0]







def  _snake_case			(							__snake_case     )  ->  Union[str, Any]:






      '''simple docstring'''







      if n <= 1:
            return n
      UpperCAmelCase_	:						Optional[Any]					   =		0
      UpperCAmelCase_	:						Dict					   =		1
      for _ in range(2       ,	n + 1     ):
            UpperCAmelCase_	:						Optional[Any]					   =		fiba, fiba + fiba
      return fiba







def  _snake_case			(							)  ->  Tuple:






      '''simple docstring'''







      for ordinal in "0th 1st 2nd 3rd 10th 100th 1000th".split():
            UpperCAmelCase_	:						List[str]					   =		int("".join(c for c in ordinal if c in "0123456789"     )     )  # 1000th --> 1000
            print(
                F"""{ordinal} fibonacci number using matrix exponentiation is """
                F"""{nth_fibonacci_matrix(__snake_case     )} and using bruteforce is """
                F"""{nth_fibonacci_bruteforce(__snake_case     )}\n"""     )
      # from timeit import timeit
      # print(timeit("nth_fibonacci_matrix(1000000)",
      #              "from main import nth_fibonacci_matrix", number=5))
      # print(timeit("nth_fibonacci_bruteforce(1000000)",
      #              "from main import nth_fibonacci_bruteforce", number=5))
      # 2.3342058970001744
      # 57.256506615000035


if __name__ == "__main__":
   import doctest

   doctest.testmod()
   main()






'''simple docstring'''





# This Python program implements an optimal binary search tree (abbreviated BST)
# building dynamic programming algorithm that delivers O(n^2) performance.
#
# The goal of the optimal BST problem is to build a low-cost BST for a
# given set of nodes, each with its own key and frequency. The frequency
# of the node is defined as how many time the node is being searched.
# The search cost of binary search tree is given by this formula:
#
# cost(1, n) = sum{i = 1 to n}((depth(node_i) + 1) * node_i_freq)
#
# where n is number of nodes in the BST. The characteristic of low-cost
# BSTs is having a faster overall search time than other implementations.
# The reason for their fast search time is that the nodes with high
# frequencies will be placed near the root of the tree while the nodes
# with low frequencies will be placed near the leaves of the tree thus
# reducing search time in the most frequent instances.
import sys
from random import randint






class     lowercase  :






      def __init__(self			,	a__			,	a__       )							-> Optional[int]:
         SCREAMING_SNAKE_CASE__      :					Union[str, Any]							    =			key
         SCREAMING_SNAKE_CASE__      :					List[Any]							    =			freq






      def __str__(self       )							-> Dict:
         return f"""Node(key={self.key}, freq={self.freq})"""


def 					UpperCamelCase						(	snake_case						,  snake_case						,  snake_case						,  snake_case						,  snake_case						,  snake_case ):




   '''simple docstring'''

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

   SCREAMING_SNAKE_CASE__      :					Tuple							    =			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(snake_case						,  snake_case						,  snake_case						,  node - 1						,  key[node]						,  snake_case )
   print_binary_search_tree(snake_case						,  snake_case						,  node + 1						,  snake_case						,  key[node]						,  snake_case )


def 					UpperCamelCase						(	snake_case ):




   '''simple docstring'''

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

   SCREAMING_SNAKE_CASE__      :					Union[str, Any]							    =			len(snake_case )

   SCREAMING_SNAKE_CASE__      :					List[str]							    =			[nodes[i].key for i in range(snake_case )]
   SCREAMING_SNAKE_CASE__      :					int							    =			[nodes[i].freq for i in range(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.
   SCREAMING_SNAKE_CASE__      :					Dict							    =			[[freqs[i] if i == j else 0 for j in range(snake_case )] for i in range(snake_case )]
   # sum[i][j] stores the sum of key frequencies between i and j inclusive in nodes
   # array
   SCREAMING_SNAKE_CASE__      :					str							    =			[[freqs[i] if i == j else 0 for j in range(snake_case )] for i in range(snake_case )]
   # stores tree roots that will be used later for constructing binary search tree
   SCREAMING_SNAKE_CASE__      :					int							    =			[[i if i == j else 0 for j in range(snake_case )] for i in range(snake_case )]

   for interval_length in range(2						,  n + 1 ):
      for i in range(n - interval_length + 1 ):
         SCREAMING_SNAKE_CASE__      :					Union[str, Any]							    =			i + interval_length - 1

         SCREAMING_SNAKE_CASE__      :					Union[str, Any]							    =			sys.maxsize  # set the value to "infinity"
         SCREAMING_SNAKE_CASE__      :					List[str]							    =			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
            SCREAMING_SNAKE_CASE__      :					Optional[Any]							    =			dp[i][r - 1] if r != i else 0  # optimal cost for left subtree
            SCREAMING_SNAKE_CASE__      :					Optional[int]							    =			dp[r + 1][j] if r != j else 0  # optimal cost for right subtree
            SCREAMING_SNAKE_CASE__      :					List[str]							    =			left + total[i][j] + right

            if dp[i][j] > cost:
               SCREAMING_SNAKE_CASE__      :					Optional[Any]							    =			cost
               SCREAMING_SNAKE_CASE__      :					str							    =			r

   print('''Binary search tree nodes:''' )
   for node in nodes:
      print(snake_case )

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


def 					UpperCamelCase						(	):




   '''simple docstring'''

   SCREAMING_SNAKE_CASE__      :					Optional[Any]							    =			[Node(snake_case						,  randint(1						,  50 ) ) for i in range(10						,  0						,  -1 )]
   find_optimal_binary_search_tree(snake_case )


if __name__ == "__main__":
       main()

