

'''simple docstring'''



from __future__ import annotations

from math import gcd


def       snake_case   (	snake_case      : int					,		snake_case      : int = 2					,		snake_case      : int = 1					,		snake_case      : int = 3					,		) ->  int | None:


    """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(snake_case      : int					,		snake_case      : int					,		snake_case      : int			) -> int:
        return (pow(snake_case					,		2			) + step) % modulus

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

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

            # 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 						=							gcd(hare - tortoise					,		snake_case			)

            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 						=							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

     _UpperCamelCase						:  Optional[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",
     )
     _UpperCamelCase						:  List[str]							       =				parser.parse_args()

     _UpperCamelCase						:  Dict							       =				pollard_rho(args.num, attempts=args.attempts)
     if divisor is None:
          print(F"""{args.num} is probably prime""")
     else:
          _UpperCamelCase						:  Optional[Any]							       =				args.num // divisor
          print(F"""{args.num} = {divisor} * {quotient}""")








'''simple docstring'''



def       snake_case   (	snake_case      : dict			) ->  set:


    """simple docstring"""


    lowerCAmelCase 						=							set()
    # edges = list of graph's edges
    lowerCAmelCase 						=							get_edges(snake_case			)

    # While there are still elements in edges list, take an arbitrary edge
    # (from_node, to_node) and add his extremity to chosen_vertices and then
    # remove all arcs adjacent to the from_node and to_node
    while edges:
        lowerCAmelCase						,   lowerCAmelCase 						=							edges.pop()
        chosen_vertices.add(snake_case			)
        chosen_vertices.add(snake_case			)
        for edge in edges.copy():
            if from_node in edge or to_node in edge:
                edges.discard(snake_case			)
    return chosen_vertices


def       snake_case   (	snake_case      : dict			) ->  set:


    """simple docstring"""


    lowerCAmelCase 						=							set()
    for from_node, to_nodes in graph.items():
        for to_node in to_nodes:
            edges.add((from_node, to_node)			)
    return edges


if __name__ == "__main__":
     import doctest

     doctest.testmod()

     # graph = {0: [1, 3], 1: [0, 3], 2: [0, 3, 4], 3: [0, 1, 2], 4: [2, 3]}
     # print(f"Matching vertex cover:\n{matching_min_vertex_cover(graph)}")





