





'''simple docstring'''






from __future__ import annotations

import math
from collections.abc import Callable





def       lowercase_      (		lowercase__				,  lowercase__				,  lowercase__				,  lowercase__ = 100				,  )   ->float:
						_snake_case:   Optional[Any]  					=		x_start
						_snake_case:   Dict  					=		fnc(lowercase__						)
						_snake_case:   int  					=		0.0

						for _ in range(lowercase__						):
												# Approximates curve as a sequence of linear lines and sums their length
												_snake_case:   List[str]  					=		(x_end - x_start) / steps + xa
												_snake_case:   Optional[Any]  					=		fnc(lowercase__						)
												length += math.hypot(xa - xa				,  fxa - fxa						)

												# Increment step
												_snake_case:   List[str]  					=		xa
												_snake_case:   Union[str, Any]  					=		fxa

						return length


if __name__ == "__main__":





		def       lowercase_      (		lowercase__						)   ->str:
								return math.sin(10 * x						)

		print('f(x) = sin(10 * x)')
		print('The length of the curve from x = -10 to x = 10 is:')
		A   : Optional[Any]             =			10
		while i <= 100_000:
				print(F'With {i} steps: {line_length(f, -10, 10, i)}')
				i *= 10












'''simple docstring'''






from __future__ import annotations

import os
from collections.abc import Mapping

A   : Optional[Any]             =			tuple[int, int]







class 			lowerCamelCase					:


					def __init__(					self	:  Tuple       ,     __snake_case	:  set[int]       ,     __snake_case	:  Mapping[EdgeT, int]       ):





											'''simple docstring'''





											_snake_case:   set[int]  					=		vertices
											_snake_case:   dict[EdgeT, int]  					=		{
											    (min(__snake_case       ), max(__snake_case       )): weight for edge, weight in edges.items()
											}


					def    SCREAMING_SNAKE_CASE_		(					self	:  int       ,     __snake_case	:  EdgeT       ,     __snake_case	:  int       ):





											'''simple docstring'''





											self.vertices.add(edge[0]       )
											self.vertices.add(edge[1]       )
											_snake_case:   Dict  					=		weight




					def    SCREAMING_SNAKE_CASE_		(					self	:  str       ):





											'''simple docstring'''





											_snake_case:   Graph  					=		Graph({min(self.vertices       )}       ,     {}       )
											_snake_case:   EdgeT
											_snake_case:   int
											_snake_case:   EdgeT
											_snake_case:   int

											while len(subgraph.vertices       ) < len(self.vertices       ):
																	_snake_case:   List[str]  					=		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:
																																			_snake_case:   Optional[Any]  					=		edge
																																			_snake_case:   Optional[int]  					=		weight

																	subgraph.add_edge(__snake_case       ,     __snake_case       )

											return subgraph





def       lowercase_      (		lowercase__ = "p107_network.txt"						)   ->int:
						_snake_case:   str  					=		os.path.abspath(os.path.dirname(lowercase__						)						)
						_snake_case:   str  					=		os.path.join(lowercase__				,  lowercase__						)
						_snake_case:   dict[EdgeT, int]  					=		{}
						_snake_case:   list[str]
						_snake_case:   int
						_snake_case:   int

						with open(lowercase__						) as f:
												_snake_case:   Tuple  					=		f.read().strip().split('\n'						)

						_snake_case:   Tuple  					=		[line.split(','						) for line in data]

						for edgea in range(1				,  len(lowercase__						)						):
												for edgea in range(lowercase__						):
																		if adjaceny_matrix[edgea][edgea] != "-":
																								_snake_case:   int  					=		int(adjaceny_matrix[edgea][edgea]						)

						_snake_case:   Graph  					=		Graph(set(range(len(lowercase__						)						)						)				,  lowercase__						)

						_snake_case:   Graph  					=		graph.prims_algorithm()

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

						return initial_total - optimal_total


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





