feat: add count_paths algorithm with a test case

graph/number_of_paths.cpp

@@ -0,0 +1,78 @@
+/**
+ * @file
+ * @brief Algorithm to count paths between two nodes in a directed graph using DFS
+ * @details
+ * This algorithm implements Depth First Search (DFS) to count the number of 
+ * possible paths between two nodes in a directed graph. It is represented using 
+ * an adjacency matrix. The algorithm recursively traverses the graph to find 
+ * all paths from the source node `u` to the destination node `v`.
+ * 
+ * @author [Aditya Borate](https://github.com/adi776borate)
+ * @see https://en.wikipedia.org/wiki/Path_(graph_theory)
+ */
+
+#include <vector>   /// for std::vector
+#include <iostream> /// for IO operations
+#include <cassert>  /// for assert
+
+/**
+ * @namespace graph
+ * @brief Graph algorithms
+ */
+namespace graph {
+
+    /**
+     * @brief Counts the number of paths from node `u` to node `v` in a directed graph
+     * using Depth First Search (DFS)
+     * 
+     * @param A adjacency matrix representing the graph (1: edge exists, 0: no edge)
+     * @param u the starting node
+     * @param v the destination node
+     * @param n the number of nodes in the graph
+     * @returns the number of paths from node `u` to node `v`
+     */
+    int count_paths(const std::vector<std::vector<int>> &A, int u, int v, int n) {
+        if (u == v) {
+            return 1; // Base case: Reached the destination node
+        }
+
+        int path_count = 0; // Count of all paths from `u` to `v`
+
+        for (int i = 0; i < n; i++) {
+            if (A[u][i] == 1) { // Check if there is an edge from `u` to `i`
+                path_count += count_paths(A, i, v, n); // Recursively explore paths from `i` to `v`
+            }
+        }
+
+        return path_count;
+    }
+
+} // namespace graph
+
+/**
+ * @brief Self-test implementations
+ * @returns void
+ */
+static void test() {
+    // Test case 1: Simple directed graph with multiple paths
+    std::vector<std::vector<int>> graph1 = {
+        {0, 1, 0, 1, 0}, 
+        {0, 0, 1, 0, 1}, 
+        {0, 0, 0, 0, 1}, 
+        {0, 0, 1, 0, 0}, 
+        {0, 0, 0, 0, 0}  
+    };
+    int n1 = 5, u1 = 0, v1 = 4;
+    assert(graph::count_paths(graph1, u1, v1, n1) == 3);  // There are 3 paths from node 0 to 4
+    std::cout << "All tests have successfully passed!\n";
+}
+
+
+/**
+ * @brief Main function
+ * @returns 0 on exit
+ */
+int main() {
+    test(); // Run self-test implementations
+    return 0;
+}