diff --git a/DIRECTORY.md b/DIRECTORY.md index d73ae11eb..4f17cf9c0 100644 --- a/DIRECTORY.md +++ b/DIRECTORY.md @@ -499,6 +499,7 @@ * [Minimum Cut](https://github.com/TheAlgorithms/Python/blob/master/networking_flow/minimum_cut.py) ## Neural Network + * [2 Hidden Layers Neural Network](https://github.com/TheAlgorithms/Python/blob/master/neural_network/2_hidden_layers_neural_network.py) * [Back Propagation Neural Network](https://github.com/TheAlgorithms/Python/blob/master/neural_network/back_propagation_neural_network.py) * [Convolution Neural Network](https://github.com/TheAlgorithms/Python/blob/master/neural_network/convolution_neural_network.py) * [Perceptron](https://github.com/TheAlgorithms/Python/blob/master/neural_network/perceptron.py) @@ -748,6 +749,8 @@ * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_101/sol1.py) * Problem 102 * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_102/sol1.py) + * Problem 107 + * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_107/sol1.py) * Problem 112 * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_112/sol1.py) * Problem 113 diff --git a/project_euler/problem_107/__init__.py b/project_euler/problem_107/__init__.py new file mode 100644 index 000000000..e69de29bb diff --git a/project_euler/problem_107/p107_network.txt b/project_euler/problem_107/p107_network.txt new file mode 100644 index 000000000..fcc3c4192 --- /dev/null +++ b/project_euler/problem_107/p107_network.txt @@ -0,0 +1,40 @@ +-,-,-,427,668,495,377,678,-,177,-,-,870,-,869,624,300,609,131,-,251,-,-,-,856,221,514,-,591,762,182,56,-,884,412,273,636,-,-,774 +-,-,262,-,-,508,472,799,-,956,578,363,940,143,-,162,122,910,-,729,802,941,922,573,531,539,667,607,-,920,-,-,315,649,937,-,185,102,636,289 +-,262,-,-,926,-,958,158,647,47,621,264,81,-,402,813,649,386,252,391,264,637,349,-,-,-,108,-,727,225,578,699,-,898,294,-,575,168,432,833 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+-,636,432,76,-,386,686,770,828,582,-,433,203,526,600,848,227,616,-,217,117,707,369,109,586,205,809,-,-,240,-,853,-,-,-,768,-,371,-,540 +774,289,833,257,-,381,239,722,711,468,933,-,-,17,-,-,148,-,-,853,-,-,-,-,264,194,260,947,-,752,147,-,-,343,112,273,344,680,540,- diff --git a/project_euler/problem_107/sol1.py b/project_euler/problem_107/sol1.py new file mode 100644 index 000000000..80a10e499 --- /dev/null +++ b/project_euler/problem_107/sol1.py @@ -0,0 +1,128 @@ +""" +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 +""" + +import os +from typing import Dict, List, Mapping, Set, Tuple + +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): + if 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) + adjacency_matrix: List[List[str]] + edges: Dict[EdgeT, int] = dict() + data: List[str] + edge1: int + edge2: int + + with open(network_file, "r") 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() = }") diff --git a/project_euler/problem_107/test_network.txt b/project_euler/problem_107/test_network.txt new file mode 100644 index 000000000..f5f2accb5 --- /dev/null +++ b/project_euler/problem_107/test_network.txt @@ -0,0 +1,7 @@ +-,16,12,21,-,-,- +16,-,-,17,20,-,- +12,-,-,28,-,31,- +21,17,28,-,18,19,23 +-,20,-,18,-,-,11 +-,-,31,19,-,-,27 +-,-,-,23,11,27,-