


from __future__ import annotations

from math import gcd






def 					lowerCamelCase_   (	_a     ,       _a = 2     ,       _a = 1     ,       _a = 3     ,       ):




		"""simple docstring"""






		if num < 2:
				raise ValueError('''The input value cannot be less than 2'''	)

		# Because of the relationship between ``f(f(x))`` and ``f(x)``, this
		# algorithm struggles to find factors that are divisible by two.
		# As a workaround, we specifically check for two and even inputs.
		#   See: https://math.stackexchange.com/a/2856214/165820
		if num > 2 and num % 2 == 0:
				return 2

		# Pollard's Rho algorithm requires a function that returns pseudorandom
		# values between 0 <= X < ``num``.  It doesn't need to be random in the
		# sense that the output value is cryptographically secure or difficult
		# to calculate, it only needs to be random in the sense that all output
		# values should be equally likely to appear.
		# For this reason, Pollard suggested using ``f(x) = (x**2 - 1) % num``
		# However, the success of Pollard's algorithm isn't guaranteed and is
		# determined in part by the initial seed and the chosen random function.
		# To make retries easier, we will instead use ``f(x) = (x**2 + C) % num``
		# where ``C`` is a value that we can modify between each attempt.
		def rand_fn(_a     ,       _a     ,       _a	) -> int:
				return (pow(_a     ,       2	) + step) % modulus

		for _ in range(_a	):
				# These track the position within the cycle detection logic.
				lowerCAmelCase__       :      List[Any]   					= seed
				lowerCAmelCase__       :      Union[str, Any]   					= seed

				while True:
						# At each iteration, the tortoise moves one step and the hare moves two.
						lowerCAmelCase__       :      str   					= rand_fn(_a     ,       _a     ,       _a	)
						lowerCAmelCase__       :      List[Any]   					= rand_fn(_a     ,       _a     ,       _a	)
						lowerCAmelCase__       :      Tuple   					= rand_fn(_a     ,       _a     ,       _a	)

						# At some point both the tortoise and the hare will enter a cycle whose
						# length ``p`` is a divisor of ``num``.  Once in that cycle, at some point
						# the tortoise and hare will end up on the same value modulo ``p``.
						# We can detect when this happens because the position difference between
						# the tortoise and the hare will share a common divisor with ``num``.
						lowerCAmelCase__       :      str   					= gcd(hare - tortoise     ,       _a	)

						if divisor == 1:
								# No common divisor yet, just keep searching.
								continue
						else:
								# We found a common divisor!
								if divisor == num:
										# Unfortunately, the divisor is ``num`` itself and is useless.
										break
								else:
										# The divisor is a nontrivial factor of ``num``!
										return divisor

        # If we made it here, then this attempt failed.
        # We need to pick a new starting seed for the tortoise and hare
        # in addition to a new step value for the random function.
        # To keep this example implementation deterministic, the
        # new values will be generated based on currently available
        # values instead of using something like ``random.randint``.

        # We can use the hare's position as the new seed.
        # This is actually what Richard Brent's the "optimized" variant does.
				lowerCAmelCase__       :      Dict   					= hare

				# The new step value for the random function can just be incremented.
				# At first the results will be similar to what the old function would
				# have produced, but the value will quickly diverge after a bit.
				step += 1

		# We haven't found a divisor within the requested number of attempts.
		# We were unlucky or ``num`` itself is actually prime.
		return None


if __name__ == "__main__":
		import argparse

		lowerCamelCase						=		argparse.ArgumentParser()
		parser.add_argument(
		    '''num''',
		    type=int,
		    help='''The value to find a divisor of''',
		)
		parser.add_argument(
		    '''--attempts''',
		    type=int,
		    default=3,
		    help='''The number of attempts before giving up''',
		)
		lowerCamelCase						=		parser.parse_args()

		lowerCamelCase						=		pollard_rho(args.num, attempts=args.attempts)
		if divisor is None:
				print(f'''{args.num} is probably prime''')
		else:
				lowerCamelCase						=		args.num // divisor
				print(f'''{args.num} = {divisor} * {quotient}''')








class  _a							:




	def __init__( self						:		str		, _SCREAMING_SNAKE_CASE						:		str		, _SCREAMING_SNAKE_CASE						:		Optional[int]		, _SCREAMING_SNAKE_CASE						:		List[Any]		)->	Tuple:
			lowerCAmelCase__       :      List[Any]   					= None
			lowerCAmelCase__       :      int   					= None
			lowerCAmelCase__       :      Union[str, Any]   					= graph

			self._normalize_graph(_SCREAMING_SNAKE_CASE		, _SCREAMING_SNAKE_CASE		)
			lowerCAmelCase__       :      Tuple   					= len(_SCREAMING_SNAKE_CASE		)
			lowerCAmelCase__       :      Optional[int]   					= None




	def 							UpperCAmelCase__( self						:		Union[str, Any]		, _SCREAMING_SNAKE_CASE						:		Optional[Any]		, _SCREAMING_SNAKE_CASE						:		Union[str, Any]		)->	Union[str, Any]:
			if sources is int:
					lowerCAmelCase__       :      List[Any]   					= [sources]
			if sinks is int:
					lowerCAmelCase__       :      Optional[Any]   					= [sinks]

			if len(_SCREAMING_SNAKE_CASE		) == 0 or len(_SCREAMING_SNAKE_CASE		) == 0:
					return

			lowerCAmelCase__       :      Union[str, Any]   					= sources[0]
			lowerCAmelCase__       :      Dict   					= sinks[0]

			# make fake vertex if there are more
			# than one source or sink
			if len(_SCREAMING_SNAKE_CASE		) > 1 or len(_SCREAMING_SNAKE_CASE		) > 1:
					lowerCAmelCase__       :      List[Any]   					= 0
					for i in sources:
							max_input_flow += sum(self.graph[i]		)

					lowerCAmelCase__       :      Any   					= len(self.graph		) + 1
					for room in self.graph:
							room.insert(0		, 0		)
					self.graph.insert(0		, [0] * size		)
					for i in sources:
							lowerCAmelCase__       :      Optional[Any]   					= max_input_flow
					lowerCAmelCase__       :      List[str]   					= 0

					lowerCAmelCase__       :      List[Any]   					= len(self.graph		) + 1
					for room in self.graph:
							room.append(0		)
					self.graph.append([0] * size		)
					for i in sinks:
							lowerCAmelCase__       :      str   					= max_input_flow
					lowerCAmelCase__       :      Tuple   					= size - 1




	def 							UpperCAmelCase__( self						:		Union[str, Any]		)->	Union[str, Any]:
			if self.maximum_flow_algorithm is None:
					raise Exception('''You need to set maximum flow algorithm before.'''		)
			if self.source_index is None or self.sink_index is None:
					return 0

			self.maximum_flow_algorithm.execute()
			return self.maximum_flow_algorithm.getMaximumFlow()




	def 							UpperCAmelCase__( self						:		str		, _SCREAMING_SNAKE_CASE						:		List[Any]		)->	int:
			lowerCAmelCase__       :      int   					= algorithm(self		)





class  _a							:




	def __init__( self						:		Optional[Any]		, _SCREAMING_SNAKE_CASE						:		Tuple		)->	Union[str, Any]:
			lowerCAmelCase__       :      Tuple   					= flow_network
			lowerCAmelCase__       :      Dict   					= flow_network.verticesCount
			lowerCAmelCase__       :      Optional[Any]   					= flow_network.sourceIndex
			lowerCAmelCase__       :      Optional[Any]   					= flow_network.sinkIndex
			# it's just a reference, so you shouldn't change
			# it in your algorithms, use deep copy before doing that
			lowerCAmelCase__       :      str   					= flow_network.graph
			lowerCAmelCase__       :      Optional[int]   					= False




	def 							UpperCAmelCase__( self						:		List[str]		)->	Dict:
			if not self.executed:
					self._algorithm()
					lowerCAmelCase__       :      Any   					= True




	def 							UpperCAmelCase__( self						:		Optional[Any]		)->	int:
			pass





class  _a							(    _lowercase):




	def __init__( self						:		Any		, _SCREAMING_SNAKE_CASE						:		List[Any]		)->	Union[str, Any]:
			super().__init__(_SCREAMING_SNAKE_CASE		)
			# use this to save your result
			lowerCAmelCase__       :      Dict   					= -1




	def 							UpperCAmelCase__( self						:		Any		)->	Optional[Any]:
			if not self.executed:
					raise Exception('''You should execute algorithm before using its result!'''		)

			return self.maximum_flow





class  _a							(    _lowercase):




	def __init__( self						:		Any		, _SCREAMING_SNAKE_CASE						:		List[str]		)->	List[str]:
			super().__init__(_SCREAMING_SNAKE_CASE		)

			lowerCAmelCase__       :      Any   					= [[0] * self.verticies_count for i in range(self.verticies_count		)]

			lowerCAmelCase__       :      Optional[Any]   					= [0] * self.verticies_count
			lowerCAmelCase__       :      str   					= [0] * self.verticies_count




	def 							UpperCAmelCase__( self						:		Any		)->	List[Any]:
			lowerCAmelCase__       :      Optional[Any]   					= self.verticies_count

			# push some substance to graph
			for nextvertex_index, bandwidth in enumerate(self.graph[self.source_index]		):
					self.preflow[self.source_index][nextvertex_index] += bandwidth
					self.preflow[nextvertex_index][self.source_index] -= bandwidth
					self.excesses[nextvertex_index] += bandwidth

			# Relabel-to-front selection rule
			lowerCAmelCase__       :      Any   					= [
			    i
			    for i in range(self.verticies_count		)
			    if i != self.source_index and i != self.sink_index
			]

			# move through list
			lowerCAmelCase__       :      str   					= 0
			while i < len(_SCREAMING_SNAKE_CASE		):
					lowerCAmelCase__       :      Union[str, Any]   					= vertices_list[i]
					lowerCAmelCase__       :      Any   					= self.heights[vertex_index]
					self.process_vertex(_SCREAMING_SNAKE_CASE		)
					if self.heights[vertex_index] > previous_height:
							# if it was relabeled, swap elements
							# and start from 0 index
							vertices_list.insert(0		, vertices_list.pop(_SCREAMING_SNAKE_CASE		)		)
							lowerCAmelCase__       :      Optional[Any]   					= 0
					else:
							i += 1

			lowerCAmelCase__       :      Optional[Any]   					= sum(self.preflow[self.source_index]		)




	def 							UpperCAmelCase__( self						:		List[str]		, _SCREAMING_SNAKE_CASE						:		Optional[Any]		)->	Optional[int]:
			while self.excesses[vertex_index] > 0:
					for neighbour_index in range(self.verticies_count		):
							# if it's neighbour and current vertex is higher
							if (
							    self.graph[vertex_index][neighbour_index]
							    - self.preflow[vertex_index][neighbour_index]
							    > 0
							    and self.heights[vertex_index] > self.heights[neighbour_index]
							):
									self.push(_SCREAMING_SNAKE_CASE		, _SCREAMING_SNAKE_CASE		)

					self.relabel(_SCREAMING_SNAKE_CASE		)




	def 							UpperCAmelCase__( self						:		int		, _SCREAMING_SNAKE_CASE						:		Optional[Any]		, _SCREAMING_SNAKE_CASE						:		Union[str, Any]		)->	Union[str, Any]:
			lowerCAmelCase__       :      Union[str, Any]   					= min(
			    self.excesses[from_index]		, self.graph[from_index][to_index] - self.preflow[from_index][to_index]		, )
			self.preflow[from_index][to_index] += preflow_delta
			self.preflow[to_index][from_index] -= preflow_delta
			self.excesses[from_index] -= preflow_delta
			self.excesses[to_index] += preflow_delta




	def 							UpperCAmelCase__( self						:		Optional[int]		, _SCREAMING_SNAKE_CASE						:		Any		)->	Optional[int]:
			lowerCAmelCase__       :      Optional[int]   					= None
			for to_index in range(self.verticies_count		):
					if (
					    self.graph[vertex_index][to_index]
					    - self.preflow[vertex_index][to_index]
					    > 0
					) and (min_height is None or self.heights[to_index] < min_height):
							lowerCAmelCase__       :      List[Any]   					= self.heights[to_index]

			if min_height is not None:
					lowerCAmelCase__       :      Optional[Any]   					= min_height + 1


if __name__ == "__main__":
		lowerCamelCase						=		[0]
		lowerCamelCase						=		[3]
		# graph = [
		#     [0, 0, 4, 6, 0, 0],
		#     [0, 0, 5, 2, 0, 0],
		#     [0, 0, 0, 0, 4, 4],
		#     [0, 0, 0, 0, 6, 6],
		#     [0, 0, 0, 0, 0, 0],
		#     [0, 0, 0, 0, 0, 0],
		# ]
		lowerCamelCase						=		[[0, 7, 0, 0], [0, 0, 6, 0], [0, 0, 0, 8], [9, 0, 0, 0]]

		# prepare our network
		lowerCamelCase						=		FlowNetwork(graph, entrances, exits)
		# set algorithm
		flow_network.set_maximum_flow_algorithm(PushRelabelExecutor)
		# and calculate
		lowerCamelCase						=		flow_network.find_maximum_flow()

		print(f'''maximum flow is {maximum_flow}''')




