
from __future__ import annotations

from math import gcd



def        UpperCamelCase    (							lowerCAmelCase__ ,			lowerCAmelCase__ = 2 ,			lowerCAmelCase__ = 1 ,			lowerCAmelCase__ = 3 ,			):
					'''simple docstring'''
					# A value less than 2 can cause an infinite loop in the algorithm.
					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(lowerCAmelCase__ ,			lowerCAmelCase__ ,			lowerCAmelCase__		) -> int:
										return (pow(lowerCAmelCase__ ,			2		) + step) % modulus

					for _ in range(lowerCAmelCase__		):
										# These track the position within the cycle detection logic.
										lowercase							=						seed
										lowercase							=						seed

										while True:
															# At each iteration, the tortoise moves one step and the hare moves two.
															lowercase							=						rand_fn(lowerCAmelCase__ ,			lowerCAmelCase__ ,			lowerCAmelCase__		)
															lowercase							=						rand_fn(lowerCAmelCase__ ,			lowerCAmelCase__ ,			lowerCAmelCase__		)
															lowercase							=						rand_fn(lowerCAmelCase__ ,			lowerCAmelCase__ ,			lowerCAmelCase__		)

															# 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``.
															lowercase							=						gcd(hare - tortoise ,			lowerCAmelCase__		)

															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.
										lowercase							=						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

							lowercase__		:Union[str, Any]   =    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",
							)
							lowercase__		:Any   =    parser.parse_args()

							lowercase__		:str   =    pollard_rho(args.num, attempts=args.attempts)
							if divisor is None:
														print(F'{args.num} is probably prime')
							else:
														lowercase__		:Union[str, Any]   =    args.num // divisor
														print(F'{args.num} = {divisor} * {quotient}')












import math

class 						lowerCAmelCase   :

					'''simple docstring'''

					def __init__(     self :		Tuple						,   __a :		int=0							)	->     Optional[Any]:  # a graph with Node 0,1,...,N-1


												"""simple docstring"""

												__lowercase					:							Any			=   n
												__lowercase					:							Optional[int]			=   [
												    [math.inf for j in range(0						,   __a							)] for i in range(0						,   __a							)
												]  # adjacency matrix for weight
												__lowercase					:							Dict			=   [
												    [math.inf for j in range(0						,   __a							)] for i in range(0						,   __a							)
												]  # dp[i][j] stores minimum distance from i to j



					def 					lowerCAmelCase (     self :		int						,   __a :		Optional[int]						,   __a :		Tuple						,   __a :		Dict							)	->     Dict:


												"""simple docstring"""

												__lowercase					:							Any			=   w



					def 					lowerCAmelCase (     self :		List[str]							)	->     Tuple:


												"""simple docstring"""

												for k in range(0						,   self.n							):
																			for i in range(0						,   self.n							):
																										for j in range(0						,   self.n							):
																																	__lowercase					:							List[str]			=   min(self.dp[i][j]						,   self.dp[i][k] + self.dp[k][j]							)



					def 					lowerCAmelCase (     self :		Any						,   __a :		Any						,   __a :		str							)	->     int:


												"""simple docstring"""

												return self.dp[u][v]


if __name__ == "__main__":
	lowerCamelCase       :       int        =				Graph(5)
	graph.add_edge(0, 2, 9)
	graph.add_edge(0, 4, 10)
	graph.add_edge(1, 3, 5)
	graph.add_edge(2, 3, 7)
	graph.add_edge(3, 0, 10)
	graph.add_edge(3, 1, 2)
	graph.add_edge(3, 2, 1)
	graph.add_edge(3, 4, 6)
	graph.add_edge(4, 1, 3)
	graph.add_edge(4, 2, 4)
	graph.add_edge(4, 3, 9)
	graph.floyd_warshall()
	graph.show_min(1, 4)
	graph.show_min(0, 3)