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| 1 | +/** |
| 2 | + * @file |
| 3 | + * @brief Algorithm to count paths between two nodes in a directed graph using DFS |
| 4 | + * @details |
| 5 | + * This algorithm implements Depth First Search (DFS) to count the number of |
| 6 | + * possible paths between two nodes in a directed graph. It is represented using |
| 7 | + * an adjacency matrix. The algorithm recursively traverses the graph to find |
| 8 | + * all paths from the source node `u` to the destination node `v`. |
| 9 | + * |
| 10 | + * @author [Aditya Borate](https://github.com/adi776borate) |
| 11 | + * @see https://en.wikipedia.org/wiki/Path_(graph_theory) |
| 12 | + */ |
| 13 | + |
| 14 | +#include <vector> /// for std::vector |
| 15 | +#include <iostream> /// for IO operations |
| 16 | +#include <cassert> /// for assert |
| 17 | +#include <cstdint> /// for fixed-size integer types (e.g., std::uint32_t) |
| 18 | + |
| 19 | +/** |
| 20 | + * @namespace graph |
| 21 | + * @brief Graph algorithms |
| 22 | + */ |
| 23 | +namespace graph { |
| 24 | + |
| 25 | + /** |
| 26 | + * @brief Helper function to perform DFS and count the number of paths from node `u` to node `v` |
| 27 | + * @param A adjacency matrix representing the graph (1: edge exists, 0: no edge) |
| 28 | + * @param u the starting node |
| 29 | + * @param v the destination node |
| 30 | + * @param n the number of nodes in the graph |
| 31 | + * @param visited a vector to keep track of visited nodes in the current DFS path |
| 32 | + * @returns the number of paths from node `u` to node `v` |
| 33 | + */ |
| 34 | + std::uint32_t count_paths_dfs(const std::vector<std::vector<std::uint32_t>>& A, |
| 35 | + std::uint32_t u, |
| 36 | + std::uint32_t v, |
| 37 | + std::uint32_t n, |
| 38 | + std::vector<bool>& visited) { |
| 39 | + if (u == v) { |
| 40 | + return 1; // Base case: Reached the destination node |
| 41 | + } |
| 42 | + |
| 43 | + visited[u] = true; // Mark the current node as visited |
| 44 | + std::uint32_t path_count = 0; // Count of all paths from `u` to `v` |
| 45 | + |
| 46 | + for (std::uint32_t i = 0; i < n; i++) { |
| 47 | + if (A[u][i] == 1 && !visited[i]) { // Check if there is an edge and the node is not visited |
| 48 | + path_count += count_paths_dfs(A, i, v, n, visited); // Recursively explore paths from `i` to `v` |
| 49 | + } |
| 50 | + } |
| 51 | + |
| 52 | + visited[u] = false; // Unmark the current node as visited (backtracking) |
| 53 | + return path_count; |
| 54 | + } |
| 55 | + |
| 56 | + |
| 57 | + /** |
| 58 | + * @brief Counts the number of paths from node `u` to node `v` in a directed graph |
| 59 | + * using Depth First Search (DFS) |
| 60 | + * |
| 61 | + * @param A adjacency matrix representing the graph (1: edge exists, 0: no edge) |
| 62 | + * @param u the starting node |
| 63 | + * @param v the destination node |
| 64 | + * @param n the number of nodes in the graph |
| 65 | + * @returns the number of paths from node `u` to node `v` |
| 66 | + */ |
| 67 | + std::uint32_t count_paths(const std::vector<std::vector<std::uint32_t>>& A, |
| 68 | + std::uint32_t u, |
| 69 | + std::uint32_t v, |
| 70 | + std::uint32_t n) { |
| 71 | + // Check for invalid nodes or empty graph |
| 72 | + if (u >= n || v >= n || A.empty() || A[0].empty()) { |
| 73 | + return 0; // No valid paths if graph is empty or nodes are out of bounds |
| 74 | + } |
| 75 | + |
| 76 | + std::vector<bool> visited(n, false); // Initialize a visited vector for tracking nodes |
| 77 | + return count_paths_dfs(A, u, v, n, visited); // Start DFS |
| 78 | + } |
| 79 | + |
| 80 | +} // namespace graph |
| 81 | + |
| 82 | +/** |
| 83 | + * @brief Self-test implementations |
| 84 | + * @returns void |
| 85 | + */ |
| 86 | +static void test() { |
| 87 | + // Test case 1: Simple directed graph with multiple paths |
| 88 | + std::vector<std::vector<std::uint32_t>> graph1 = { |
| 89 | + {0, 1, 0, 1, 0}, |
| 90 | + {0, 0, 1, 0, 1}, |
| 91 | + {0, 0, 0, 0, 1}, |
| 92 | + {0, 0, 1, 0, 0}, |
| 93 | + {0, 0, 0, 0, 0} |
| 94 | + }; |
| 95 | + std::uint32_t n1 = 5, u1 = 0, v1 = 4; |
| 96 | + assert(graph::count_paths(graph1, u1, v1, n1) == 3); // There are 3 paths from node 0 to 4 |
| 97 | + |
| 98 | + // Test case 2: No possible path (disconnected graph) |
| 99 | + std::vector<std::vector<std::uint32_t>> graph2 = { |
| 100 | + {0, 1, 0, 0, 0}, |
| 101 | + {0, 0, 0, 0, 0}, |
| 102 | + {0, 0, 0, 0, 1}, |
| 103 | + {0, 0, 1, 0, 0}, |
| 104 | + {0, 0, 0, 0, 0} |
| 105 | + }; |
| 106 | + std::uint32_t n2 = 5, u2 = 0, v2 = 4; |
| 107 | + assert(graph::count_paths(graph2, u2, v2, n2) == 0); // No path from node 0 to 4 |
| 108 | + |
| 109 | + // Test case 3: Cyclic graph with multiple paths |
| 110 | + std::vector<std::vector<std::uint32_t>> graph3 = { |
| 111 | + {0, 1, 0, 0, 0}, |
| 112 | + {0, 0, 1, 1, 0}, |
| 113 | + {1, 0, 0, 0, 1}, |
| 114 | + {0, 0, 1, 0, 1}, |
| 115 | + {0, 0, 0, 0, 0} |
| 116 | + }; |
| 117 | + std::uint32_t n3 = 5, u3 = 0, v3 = 4; |
| 118 | + assert(graph::count_paths(graph3, u3, v3, n3) == 3); // There are 3 paths from node 0 to 4 |
| 119 | + |
| 120 | + // Test case 4: Single node graph (self-loop) |
| 121 | + std::vector<std::vector<std::uint32_t>> graph4 = { |
| 122 | + {0} |
| 123 | + }; |
| 124 | + std::uint32_t n4 = 1, u4 = 0, v4 = 0; |
| 125 | + assert(graph::count_paths(graph4, u4, v4, n4) == 1); // There is self-loop, so 1 path from node 0 to 0 |
| 126 | + |
| 127 | + // Test case 5: Empty graph (no nodes, no paths) |
| 128 | + std::vector<std::vector<std::uint32_t>> graph5 = {{}}; |
| 129 | + int n5 = 0, u5 = 0, v5 = 0; |
| 130 | + assert(graph::count_paths(graph5, u5, v5, n5) == 0); // There are no paths in an empty graph |
| 131 | + |
| 132 | + // Test case 6: Invalid nodes (out of bounds) |
| 133 | + std::vector<std::vector<std::uint32_t>> graph6 = { |
| 134 | + {0, 1, 0}, |
| 135 | + {0, 0, 1}, |
| 136 | + {0, 0, 0} |
| 137 | + }; |
| 138 | + int n6 = 3, u6 = 0, v6 = 5; // Node `v` is out of bounds (n = 3, so valid indices are 0, 1, 2) |
| 139 | + assert(graph::count_paths(graph6, u6, v6, n6) == 0); // Should return 0 because `v = 5` is invalid |
| 140 | + |
| 141 | + std::cout << "All tests have successfully passed!\n"; |
| 142 | +} |
| 143 | + |
| 144 | +/** |
| 145 | + * @brief Main function |
| 146 | + * @returns 0 on exit |
| 147 | + */ |
| 148 | +int main() { |
| 149 | + test(); // Run self-test implementations |
| 150 | + return 0; |
| 151 | +} |
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