Implement Parentheses Generator using Backtracking Strategy (#713)

* chore: add `parentheses_generator.rs` to DIRECTORY.md

* feat: implement Parentheses Generator algorithm

* chore: rename `open` to `open_count`, `close` to `close_count`

* chore: change `u32` to `usize`

* ref: eliminate the need for mutability

* chore: handle case where `n = 0`

* chore: fix typos in docstring

* style: simplify logic

---------

Co-authored-by: Piotr Idzik <[email protected]>

Author: sozelfist

DIRECTORY.md

@@ -5,6 +5,7 @@
     * [All Combination Of Size K](https://github.com/TheAlgorithms/Rust/blob/master/src/backtracking/all_combination_of_size_k.rs)
     * [Knight Tour](https://github.com/TheAlgorithms/Rust/blob/master/src/backtracking/knight_tour.rs)
     * [N Queens](https://github.com/TheAlgorithms/Rust/blob/master/src/backtracking/n_queens.rs)
+    * [Parentheses Generator](https://github.com/TheAlgorithms/Rust/blob/master/src/backtracking/parentheses_generator.rs)
     * [Permutations](https://github.com/TheAlgorithms/Rust/blob/master/src/backtracking/permutations.rs)
     * [Sudoku](https://github.com/TheAlgorithms/Rust/blob/master/src/backtracking/sudoku.rs)
   * Big Integer

src/backtracking/mod.rs

@@ -1,11 +1,13 @@
 mod all_combination_of_size_k;
 mod knight_tour;
 mod n_queens;
+mod parentheses_generator;
 mod permutations;
 mod sudoku;
 
 pub use all_combination_of_size_k::generate_all_combinations;
 pub use knight_tour::find_knight_tour;
 pub use n_queens::n_queens_solver;
+pub use parentheses_generator::generate_parentheses;
 pub use permutations::permute;
 pub use sudoku::sudoku_solver;

src/backtracking/parentheses_generator.rs

@@ -0,0 +1,76 @@
+/// Generates all combinations of well-formed parentheses given a non-negative integer `n`.
+///
+/// This function uses backtracking to generate all possible combinations of well-formed
+/// parentheses. The resulting combinations are returned as a vector of strings.
+///
+/// # Arguments
+///
+/// * `n` - A non-negative integer representing the number of pairs of parentheses.
+pub fn generate_parentheses(n: usize) -> Vec<String> {
+    let mut result = Vec::new();
+    if n > 0 {
+        generate("", 0, 0, n, &mut result);
+    }
+    result
+}
+
+/// Helper function for generating parentheses recursively.
+///
+/// This function is called recursively to build combinations of well-formed parentheses.
+/// It tracks the number of open and close parentheses added so far and adds a new parenthesis
+/// if it's valid to do so.
+///
+/// # Arguments
+///
+/// * `current` - The current string of parentheses being built.
+/// * `open_count` - The count of open parentheses in the current string.
+/// * `close_count` - The count of close parentheses in the current string.
+/// * `n` - The total number of pairs of parentheses to be generated.
+/// * `result` - A mutable reference to the vector storing the generated combinations.
+fn generate(
+    current: &str,
+    open_count: usize,
+    close_count: usize,
+    n: usize,
+    result: &mut Vec<String>,
+) {
+    if current.len() == (n * 2) {
+        result.push(current.to_string());
+        return;
+    }
+
+    if open_count < n {
+        let new_str = current.to_string() + "(";
+        generate(&new_str, open_count + 1, close_count, n, result);
+    }
+
+    if close_count < open_count {
+        let new_str = current.to_string() + ")";
+        generate(&new_str, open_count, close_count + 1, n, result);
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    macro_rules! generate_parentheses_tests {
+        ($($name:ident: $test_case:expr,)*) => {
+            $(
+                #[test]
+                fn $name() {
+                    let (n, expected_result) = $test_case;
+                    assert_eq!(generate_parentheses(n), expected_result);
+                }
+            )*
+        };
+    }
+
+    generate_parentheses_tests! {
+        test_generate_parentheses_0: (0, Vec::<String>::new()),
+        test_generate_parentheses_1: (1, vec!["()"]),
+        test_generate_parentheses_2: (2, vec!["(())", "()()"]),
+        test_generate_parentheses_3: (3, vec!["((()))", "(()())", "(())()", "()(())", "()()()"]),
+        test_generate_parentheses_4: (4, vec!["(((())))", "((()()))", "((())())", "((()))()", "(()(()))", "(()()())", "(()())()", "(())(())", "(())()()", "()((()))", "()(()())", "()(())()", "()()(())", "()()()()"]),
+    }
+}