






def   _A   (				SCREAMING_SNAKE_CASE__  :	int     ,  SCREAMING_SNAKE_CASE__  :	list       ):
    _enforce_args(UpperCamelCase__     ,  UpperCamelCase__       )
    if n == 0:
        return 0
    UpperCamelCase       :List[str]								=   float('''-inf'''       )
    for i in range(1     ,  n + 1       ):
        UpperCamelCase       :List[Any]								=   max(
            UpperCamelCase__     ,  prices[i - 1] + naive_cut_rod_recursive(n - i     ,  UpperCamelCase__       )       )

    return max_revue


def   _A   (				SCREAMING_SNAKE_CASE__  :	int     ,  SCREAMING_SNAKE_CASE__  :	list       ):
    _enforce_args(UpperCamelCase__     ,  UpperCamelCase__       )
    UpperCamelCase       :int								=   [float('''-inf'''       ) for _ in range(n + 1       )]
    return _top_down_cut_rod_recursive(UpperCamelCase__     ,  UpperCamelCase__     ,  UpperCamelCase__       )


def   _A   (				SCREAMING_SNAKE_CASE__  :	int     ,  SCREAMING_SNAKE_CASE__  :	list     ,  SCREAMING_SNAKE_CASE__  :	list       ):
    if max_rev[n] >= 0:
        return max_rev[n]
    elif n == 0:
        return 0
    else:
        UpperCamelCase       :str								=   float('''-inf'''       )
        for i in range(1     ,  n + 1       ):
            UpperCamelCase       :List[str]								=   max(
                UpperCamelCase__     ,  prices[i - 1] + _top_down_cut_rod_recursive(n - i     ,  UpperCamelCase__     ,  UpperCamelCase__       )     ,  )

        UpperCamelCase       :Union[str, Any]								=   max_revenue

    return max_rev[n]


def   _A   (				SCREAMING_SNAKE_CASE__  :	int     ,  SCREAMING_SNAKE_CASE__  :	list       ):
    _enforce_args(UpperCamelCase__     ,  UpperCamelCase__       )

    # length(max_rev) = n + 1, to accommodate for the revenue obtainable from a rod of
    # length 0.
    UpperCamelCase       :Any								=   [float('''-inf'''       ) for _ in range(n + 1       )]
    UpperCamelCase       :int								=   0

    for i in range(1     ,  n + 1       ):
        UpperCamelCase       :Dict								=   max_rev[i]
        for j in range(1     ,  i + 1       ):
            UpperCamelCase       :str								=   max(UpperCamelCase__     ,  prices[j - 1] + max_rev[i - j]       )

        UpperCamelCase       :Optional[int]								=   max_revenue_i

    return max_rev[n]


def   _A   (				SCREAMING_SNAKE_CASE__  :	int     ,  SCREAMING_SNAKE_CASE__  :	list       ):
    if n < 0:
        UpperCamelCase       :Optional[int]								=   F'''n must be greater than or equal to 0. Got n = {n}'''
        raise ValueError(UpperCamelCase__       )

    if n > len(UpperCamelCase__       ):
        UpperCamelCase       :int								=   (
            '''Each integral piece of rod must have a corresponding price. '''
            F'''Got n = {n} but length of prices = {len(UpperCamelCase__       )}'''
        )
        raise ValueError(UpperCamelCase__       )


def   _A   (				):
    UpperCamelCase       :Union[str, Any]								=   [6, 10, 12, 15, 20, 23]
    UpperCamelCase       :str								=   len(UpperCamelCase__       )

    # the best revenue comes from cutting the rod into 6 pieces, each
    # of length 1 resulting in a revenue of 6 * 6 = 36.
    UpperCamelCase       :List[str]								=   36

    UpperCamelCase       :Dict								=   top_down_cut_rod(UpperCamelCase__     ,  UpperCamelCase__       )
    UpperCamelCase       :Optional[Any]								=   bottom_up_cut_rod(UpperCamelCase__     ,  UpperCamelCase__       )
    UpperCamelCase       :List[str]								=   naive_cut_rod_recursive(UpperCamelCase__     ,  UpperCamelCase__       )

    assert expected_max_revenue == max_rev_top_down
    assert max_rev_top_down == max_rev_bottom_up
    assert max_rev_bottom_up == max_rev_naive


if __name__ == "__main__":
 main()










"""simple docstring"""
from __future__ import annotations

import os
from collections.abc import Mapping

_lowerCAmelCase    :Tuple     =					tuple[int, int]
class 							_UpperCAmelCase							:





     '''simple docstring'''



     def __init__(					self       ,	A       ,	A		)							->					None:
         _UpperCAmelCase							:  set[int]											=    vertices
         _UpperCAmelCase							:  dict[EdgeT, int]											=    {
             (min(A		), max(A		)): weight for edge, weight in edges.items()
         }





     def  __lowerCAmelCase	(					self       ,	A       ,	A		)							->					None:
         self.vertices.add(edge[0]		)
         self.vertices.add(edge[1]		)
         _UpperCAmelCase							:  List[Any]											=    weight





     def  __lowerCAmelCase	(					self		)							->					Graph:
         _UpperCAmelCase							:  Graph											=    Graph({min(self.vertices		)}       ,	{}		)
         _UpperCAmelCase							:  EdgeT
         _UpperCAmelCase							:  int
         _UpperCAmelCase							:  EdgeT
         _UpperCAmelCase							:  int

         while len(subgraph.vertices		) < len(self.vertices		):
             _UpperCAmelCase							:  Any											=    max(self.edges.values()		) + 1
             for edge, weight in self.edges.items():
                 if (edge[0] in subgraph.vertices) ^ (edge[1] in subgraph.vertices):
                     if weight < min_weight:
                         _UpperCAmelCase							:  Tuple											=    edge
                         _UpperCAmelCase							:  Optional[int]											=    weight

             subgraph.add_edge(A       ,	A		)

         return subgraph





def  lowerCamelCase_						(UpperCamelCase__		:		str = "p107_network.txt"							):
    _UpperCAmelCase							:  str											=    os.path.abspath(os.path.dirname(UpperCamelCase__							)							)
    _UpperCAmelCase							:  str											=    os.path.join(UpperCamelCase__     ,       UpperCamelCase__							)
    _UpperCAmelCase							:  dict[EdgeT, int]											=    {}
    _UpperCAmelCase							:  list[str]
    _UpperCAmelCase							:  int
    _UpperCAmelCase							:  int

    with open(UpperCamelCase__							) as f:
        _UpperCAmelCase							:  str											=    f.read().strip().split('''\n'''							)

    _UpperCAmelCase							:  List[Any]											=    [line.split(''','''							) for line in data]

    for edgea in range(1     ,       len(UpperCamelCase__							)							):
        for edgea in range(UpperCamelCase__							):
            if adjaceny_matrix[edgea][edgea] != "-":
                _UpperCAmelCase							:  Optional[Any]											=    int(adjaceny_matrix[edgea][edgea]							)

    _UpperCAmelCase							:  Graph											=    Graph(set(range(len(UpperCamelCase__							)							)							)     ,       UpperCamelCase__							)

    _UpperCAmelCase							:  Graph											=    graph.prims_algorithm()

    _UpperCAmelCase							:  int											=    sum(graph.edges.values()							)
    _UpperCAmelCase							:  int											=    sum(subgraph.edges.values()							)

    return initial_total - optimal_total


if __name__ == "__main__":
  print(f"{solution() = }")


