




'''simple docstring'''





from __future__ import annotations

from collections import deque
from queue import Queue
from timeit import timeit

lowercase__								=						{
    '''A''': ['''B''', '''C'''],
    '''B''': ['''A''', '''D''', '''E'''],
    '''C''': ['''A''', '''F'''],
    '''D''': ['''B'''],
    '''E''': ['''B''', '''F'''],
    '''F''': ['''C''', '''E'''],
}


def 					UpperCamelCase						(	snake_case						,  snake_case ):




   '''simple docstring'''

   SCREAMING_SNAKE_CASE__      :					Tuple							    =			{start}
   SCREAMING_SNAKE_CASE__      :					Any							    =			[start]
   SCREAMING_SNAKE_CASE__      :					Queue							    =			Queue()
   queue.put(snake_case )
   while not queue.empty():
      SCREAMING_SNAKE_CASE__      :					Union[str, Any]							    =			queue.get()
      for w in graph[v]:
         if w not in explored:
            explored.add(snake_case )
            result.append(snake_case )
            queue.put(snake_case )
   return result


def 					UpperCamelCase						(	snake_case						,  snake_case ):




   '''simple docstring'''

   SCREAMING_SNAKE_CASE__      :					int							    =			{start}
   SCREAMING_SNAKE_CASE__      :					Optional[int]							    =			[start]
   SCREAMING_SNAKE_CASE__      :					Tuple							    =			deque([start] )
   while queue:
      SCREAMING_SNAKE_CASE__      :					List[str]							    =			queue.popleft()
      for child in graph[v]:
         if child not in visited:
            visited.add(snake_case )
            result.append(snake_case )
            queue.append(snake_case )
   return result


def 					UpperCamelCase						(	snake_case ):




   '''simple docstring'''

   SCREAMING_SNAKE_CASE__      :					List[str]							    =			f"""from __main__ import G, {name}"""
   SCREAMING_SNAKE_CASE__      :					List[str]							    =			1_00_00
   SCREAMING_SNAKE_CASE__      :					Any							    =			timeit(f"""{name}(G, 'A')"""						,  setup=snake_case						,  number=snake_case )
   print(f"""{name:<35} finished {number} runs in {res:.5f} seconds""" )


if __name__ == "__main__":
       import doctest

       doctest.testmod()

       benchmark_function('''breadth_first_search''')
       benchmark_function('''breadth_first_search_with_deque''')
       # breadth_first_search                finished 10000 runs in 0.20999 seconds
       # breadth_first_search_with_deque     finished 10000 runs in 0.01421 seconds







'''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()

