feat: add well-formed parentheses generator

DIRECTORY.md

@@ -1,5 +1,6 @@
 
 ## Backtracking
+  * [Generate Parentheses](https://github.com/TheAlgorithms/C-Plus-Plus/blob/HEAD/backtracking/generate_parentheses.cpp)
   * [Graph Coloring](https://github.com/TheAlgorithms/C-Plus-Plus/blob/HEAD/backtracking/graph_coloring.cpp)
   * [Knight Tour](https://github.com/TheAlgorithms/C-Plus-Plus/blob/HEAD/backtracking/knight_tour.cpp)
   * [Magic Sequence](https://github.com/TheAlgorithms/C-Plus-Plus/blob/HEAD/backtracking/magic_sequence.cpp)

backtracking/generate_parentheses.cpp

@@ -0,0 +1,97 @@
+/**
+ * @file
+ * @brief Generates all combinations of well-formed parentheses.
+ *
+ * @details a sequence of parentheses is well-formed if each opening parentheses
+ has a corresponding closing parenthesis
+ *  and the closing parentheses are correctly ordered
+ *
+ * @author [Giuseppe Coco](https://github.com/WoWS17)
+
+ */
+
+#include <cassert>
+#include <iostream>
+#include <vector>
+
+class generate_parentheses {
+ private:
+    std::vector<std::string> res;  // Contains all possible valid patterns
+
+    /**
+     * @brief function that implements backtracking
+     *
+     * @param str string build during backtracking
+     * @param n number of pairs of parentheses
+     * @param closed number of closed parentheses
+     * @param open number of open parentheses
+     */
+
+    void makeStrings(std::string str, int n, int closed, int open) {
+        if (closed > open)  // We can never have more closed than open
+            return;
+
+        if (str.length() == 2 * n and
+            closed != open)  // closed and open must be the same
+            return;
+
+        if (str.length() == 2 * n) {
+            res.push_back(str);
+            return;
+        }
+
+        makeStrings(str + ')', n, closed + 1, open);
+        makeStrings(str + '(', n, closed, open + 1);
+    }
+
+ public:
+    /**
+     * @brief wrapper interface
+     *
+     * @param n number of pairs of parentheses
+     * @return all well-formed pattern of parentheses
+     */
+    std::vector<std::string> generate_parenthesis(int n) {
+        res.clear();
+        std::string str = "(";
+        makeStrings(str, n, 0, 1);
+        return res;
+    }
+};
+
+/**
+ * @brief Self-test implementations
+ * @returns void
+ */
+static void test() {
+    int n;
+    std::vector<std::string> patterns;
+    generate_parentheses p;
+
+    n = 1;
+    patterns = {{"()"}};
+    assert(p.generate_parenthesis(n) == patterns);
+
+    n = 3;
+    patterns = {{"()()()"}, {"()(())"}, {"(())()"}, {"(()())"}, {"((()))"}};
+
+    assert(p.generate_parenthesis(n) == patterns);
+
+    n = 4;
+    patterns = {{"()()()()"}, {"()()(())"}, {"()(())()"}, {"()(()())"},
+                {"()((()))"}, {"(())()()"}, {"(())(())"}, {"(()())()"},
+                {"(()()())"}, {"(()(()))"}, {"((()))()"}, {"((())())"},
+                {"((()()))"}, {"(((())))"}};
+    assert(p.generate_parenthesis(n) == patterns);
+
+    std::cout << "All tests passed\n";
+}
+
+/**
+ * @brief Main function
+ * @returns 0 on exit
+ */
+int main() {
+    test();
+    return 0;
+}