

_a:      Optional[Any]								=     float("""inf""")

class   __UpperCamelCase      :




	def __init__(						self		:  List[Any]						,		lowerCAmelCase		:  Optional[int]							):


		'''simple docstring'''







		UpperCAmelCase_    			=     [0] * n
		UpperCAmelCase_    			=     [0] * n
		UpperCAmelCase_    			=     [0] * n
		UpperCAmelCase_    			=     [[] for _ in range(snake_case_							)]




	def  __A   (						self		:  Any						,		lowerCAmelCase		:  Tuple						,		lowerCAmelCase		:  str						,		lowerCAmelCase		:  str						,		lowerCAmelCase		:  List[Any]=0							):


		'''simple docstring'''







		self.adj[a].append([b, len(self.adj[b]							), c, 0]							)
		self.adj[b].append([a, len(self.adj[a]							) - 1, rcap, 0]							)




	def  __A   (						self		:  Dict						,		lowerCAmelCase		:  Optional[Any]						,		lowerCAmelCase		:  Dict						,		lowerCAmelCase		:  Union[str, Any]							):


		'''simple docstring'''







		if vertex == sink or not flow:
			return flow

		for i in range(self.ptr[vertex]						,		len(self.adj[vertex]							)							):
			UpperCAmelCase_    			=     self.adj[vertex][i]
			if self.lvl[e[0]] == self.lvl[vertex] + 1:
				UpperCAmelCase_    			=     self.depth_first_search(e[0]						,		snake_case_						,		min(snake_case_						,		e[2] - e[3]							)							)
				if p:
					self.adj[vertex][i][3] += p
					self.adj[e[0]][e[1]][3] -= p
					return p
			UpperCAmelCase_    			=     self.ptr[vertex] + 1
		return 0




	def  __A   (						self		:  str						,		lowerCAmelCase		:  Tuple						,		lowerCAmelCase		:  List[str]							):


		'''simple docstring'''







		UpperCAmelCase_			,				UpperCAmelCase_    			=     0, source
		for l in range(31							):  # noqa: E741  l = 30 maybe faster for random data
			while True:
				UpperCAmelCase_			,				UpperCAmelCase_    			=     [0] * len(self.q							), [0] * len(self.q							)
				UpperCAmelCase_			,				UpperCAmelCase_			,				UpperCAmelCase_    			=     0, 1, 1
				while qi < qe and not self.lvl[sink]:
					UpperCAmelCase_    			=     self.q[qi]
					qi += 1
					for e in self.adj[v]:
						if not self.lvl[e[0]] and (e[2] - e[3]) >> (30 - l):
							UpperCAmelCase_    			=     e[0]
							qe += 1
							UpperCAmelCase_    			=     self.lvl[v] + 1

				UpperCAmelCase_    			=     self.depth_first_search(snake_case_						,		snake_case_						,		snake_case_							)
				while p:
					flow += p
					UpperCAmelCase_    			=     self.depth_first_search(snake_case_						,		snake_case_						,		snake_case_							)

				if not self.lvl[sink]:
					break

		return flow


_a:      Any								=     Dinic(10)
_a:      List[str]								=     0
_a:      Any								=     9
for vertex in range(1, 5):
				graph.add_edge(source, vertex, 1)
for vertex in range(5, 9):
				graph.add_edge(vertex, sink, 1)
for vertex in range(1, 5):
				graph.add_edge(vertex, vertex + 4, 1)

# Now we can know that is the maximum flow(source -> sink)
print(graph.max_flow(source, sink))


# 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   __UpperCamelCase      :




	def __init__(						self		:  Optional[Any]						,		lowerCAmelCase		:  Any						,		lowerCAmelCase		:  Optional[Any]							):


		'''simple docstring'''







		UpperCAmelCase_    			=     key
		UpperCAmelCase_    			=     freq




	def __str__(						self		:  List[str]							):


		'''simple docstring'''







		return F"Node(key={self.key}, freq={self.freq})"



def  __lowerCAmelCase (  A    ,       A    ,       A    ,       A    ,       A    ,       A      ):
	if i > j or i < 0 or j > len(A      ) - 1:
		return

	UpperCAmelCase_    			=     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(A    ,       A    ,       A    ,       node - 1    ,       key[node]    ,       A      )
	print_binary_search_tree(A    ,       A    ,       node + 1    ,       A    ,       key[node]    ,       A      )



def  __lowerCAmelCase (  A      ):
	nodes.sort(key=lambda A      : node.key      )

	UpperCAmelCase_    			=     len(A      )

	UpperCAmelCase_    			=     [nodes[i].key for i in range(A      )]
	UpperCAmelCase_    			=     [nodes[i].freq for i in range(A      )]

	# 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.
	UpperCAmelCase_    			=     [[freqs[i] if i == j else 0 for j in range(A      )] for i in range(A      )]
	# sum[i][j] stores the sum of key frequencies between i and j inclusive in nodes
	# array
	UpperCAmelCase_    			=     [[freqs[i] if i == j else 0 for j in range(A      )] for i in range(A      )]
	# stores tree roots that will be used later for constructing binary search tree
	UpperCAmelCase_    			=     [[i if i == j else 0 for j in range(A      )] for i in range(A      )]

	for interval_length in range(2    ,       n + 1      ):
		for i in range(n - interval_length + 1      ):
			UpperCAmelCase_    			=     i + interval_length - 1

			UpperCAmelCase_    			=     sys.maxsize  # set the value to "infinity"
			UpperCAmelCase_    			=     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
				UpperCAmelCase_    			=     dp[i][r - 1] if r != i else 0  # optimal cost for left subtree
				UpperCAmelCase_    			=     dp[r + 1][j] if r != j else 0  # optimal cost for right subtree
				UpperCAmelCase_    			=     left + total[i][j] + right

				if dp[i][j] > cost:
					UpperCAmelCase_    			=     cost
					UpperCAmelCase_    			=     r

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

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



def  __lowerCAmelCase (  ):
	# A sample binary search tree
	UpperCAmelCase_    			=     [Node(A    ,       randint(1    ,       50      )      ) for i in range(10    ,       0    ,       -1      )]
	find_optimal_binary_search_tree(A      )


if __name__ == "__main__":
				main()