mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-11-23 21:11:08 +00:00
93fb555e0a
* Enable ruff SIM102 rule * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * Fix * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --------- Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
131 lines
4.1 KiB
Python
131 lines
4.1 KiB
Python
"""
|
||
The following undirected network consists of seven vertices and twelve edges
|
||
with a total weight of 243.
|
||

|
||
The same network can be represented by the matrix below.
|
||
|
||
A B C D E F G
|
||
A - 16 12 21 - - -
|
||
B 16 - - 17 20 - -
|
||
C 12 - - 28 - 31 -
|
||
D 21 17 28 - 18 19 23
|
||
E - 20 - 18 - - 11
|
||
F - - 31 19 - - 27
|
||
G - - - 23 11 27 -
|
||
|
||
However, it is possible to optimise the network by removing some edges and still
|
||
ensure that all points on the network remain connected. The network which achieves
|
||
the maximum saving is shown below. It has a weight of 93, representing a saving of
|
||
243 - 93 = 150 from the original network.
|
||
|
||
Using network.txt (right click and 'Save Link/Target As...'), a 6K text file
|
||
containing a network with forty vertices, and given in matrix form, find the maximum
|
||
saving which can be achieved by removing redundant edges whilst ensuring that the
|
||
network remains connected.
|
||
|
||
Solution:
|
||
We use Prim's algorithm to find a Minimum Spanning Tree.
|
||
Reference: https://en.wikipedia.org/wiki/Prim%27s_algorithm
|
||
"""
|
||
|
||
from __future__ import annotations
|
||
|
||
import os
|
||
from collections.abc import Mapping
|
||
|
||
EdgeT = tuple[int, int]
|
||
|
||
|
||
class Graph:
|
||
"""
|
||
A class representing an undirected weighted graph.
|
||
"""
|
||
|
||
def __init__(self, vertices: set[int], edges: Mapping[EdgeT, int]) -> None:
|
||
self.vertices: set[int] = vertices
|
||
self.edges: dict[EdgeT, int] = {
|
||
(min(edge), max(edge)): weight for edge, weight in edges.items()
|
||
}
|
||
|
||
def add_edge(self, edge: EdgeT, weight: int) -> None:
|
||
"""
|
||
Add a new edge to the graph.
|
||
>>> graph = Graph({1, 2}, {(2, 1): 4})
|
||
>>> graph.add_edge((3, 1), 5)
|
||
>>> sorted(graph.vertices)
|
||
[1, 2, 3]
|
||
>>> sorted([(v,k) for k,v in graph.edges.items()])
|
||
[(4, (1, 2)), (5, (1, 3))]
|
||
"""
|
||
self.vertices.add(edge[0])
|
||
self.vertices.add(edge[1])
|
||
self.edges[(min(edge), max(edge))] = weight
|
||
|
||
def prims_algorithm(self) -> Graph:
|
||
"""
|
||
Run Prim's algorithm to find the minimum spanning tree.
|
||
Reference: https://en.wikipedia.org/wiki/Prim%27s_algorithm
|
||
>>> graph = Graph({1,2,3,4},{(1,2):5, (1,3):10, (1,4):20, (2,4):30, (3,4):1})
|
||
>>> mst = graph.prims_algorithm()
|
||
>>> sorted(mst.vertices)
|
||
[1, 2, 3, 4]
|
||
>>> sorted(mst.edges)
|
||
[(1, 2), (1, 3), (3, 4)]
|
||
"""
|
||
subgraph: Graph = Graph({min(self.vertices)}, {})
|
||
min_edge: EdgeT
|
||
min_weight: int
|
||
edge: EdgeT
|
||
weight: int
|
||
|
||
while len(subgraph.vertices) < len(self.vertices):
|
||
min_weight = max(self.edges.values()) + 1
|
||
for edge, weight in self.edges.items():
|
||
if (edge[0] in subgraph.vertices) ^ (
|
||
edge[1] in subgraph.vertices
|
||
) and weight < min_weight:
|
||
min_edge = edge
|
||
min_weight = weight
|
||
|
||
subgraph.add_edge(min_edge, min_weight)
|
||
|
||
return subgraph
|
||
|
||
|
||
def solution(filename: str = "p107_network.txt") -> int:
|
||
"""
|
||
Find the maximum saving which can be achieved by removing redundant edges
|
||
whilst ensuring that the network remains connected.
|
||
>>> solution("test_network.txt")
|
||
150
|
||
"""
|
||
script_dir: str = os.path.abspath(os.path.dirname(__file__))
|
||
network_file: str = os.path.join(script_dir, filename)
|
||
edges: dict[EdgeT, int] = {}
|
||
data: list[str]
|
||
edge1: int
|
||
edge2: int
|
||
|
||
with open(network_file) as f:
|
||
data = f.read().strip().split("\n")
|
||
|
||
adjaceny_matrix = [line.split(",") for line in data]
|
||
|
||
for edge1 in range(1, len(adjaceny_matrix)):
|
||
for edge2 in range(edge1):
|
||
if adjaceny_matrix[edge1][edge2] != "-":
|
||
edges[(edge2, edge1)] = int(adjaceny_matrix[edge1][edge2])
|
||
|
||
graph: Graph = Graph(set(range(len(adjaceny_matrix))), edges)
|
||
|
||
subgraph: Graph = graph.prims_algorithm()
|
||
|
||
initial_total: int = sum(graph.edges.values())
|
||
optimal_total: int = sum(subgraph.edges.values())
|
||
|
||
return initial_total - optimal_total
|
||
|
||
|
||
if __name__ == "__main__":
|
||
print(f"{solution() = }")
|