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