Add solution for Project Euler problem 107 (TheAlgorithms#4066)

* Added solution for Project Euler problem 107

* Doctests and better variable names

* Type hints

* Small edits

* Forward reference for typing hint

* updating DIRECTORY.md

Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>

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

project_euler/problem_107/p107_network.txt

@@ -0,0 +1,40 @@
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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() = }")

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,-