Add function documentation in Sudoku.java

DIRECTORY.md

@@ -238,6 +238,7 @@
             * [KnapsackMemoization](https://github.com/TheAlgorithms/Java/blob/master/src/main/java/com/thealgorithms/dynamicprogramming/KnapsackMemoization.java)
             * [LevenshteinDistance](https://github.com/TheAlgorithms/Java/blob/master/src/main/java/com/thealgorithms/dynamicprogramming/LevenshteinDistance.java)
             * [LongestAlternatingSubsequence](https://github.com/TheAlgorithms/Java/blob/master/src/main/java/com/thealgorithms/dynamicprogramming/LongestAlternatingSubsequence.java)
+            * [LongestArithmeticSubsequence](https://github.com/TheAlgorithms/Java/blob/master/src/main/java/com/thealgorithms/dynamicprogramming/LongestArithmeticSubsequence.java)
             * [LongestCommonSubsequence](https://github.com/TheAlgorithms/Java/blob/master/src/main/java/com/thealgorithms/dynamicprogramming/LongestCommonSubsequence.java)
             * [LongestIncreasingSubsequence](https://github.com/TheAlgorithms/Java/blob/master/src/main/java/com/thealgorithms/dynamicprogramming/LongestIncreasingSubsequence.java)
             * [LongestPalindromicSubsequence](https://github.com/TheAlgorithms/Java/blob/master/src/main/java/com/thealgorithms/dynamicprogramming/LongestPalindromicSubsequence.java)
@@ -733,6 +734,7 @@
             * [KnapsackTest](https://github.com/TheAlgorithms/Java/blob/master/src/test/java/com/thealgorithms/dynamicprogramming/KnapsackTest.java)
             * [LevenshteinDistanceTests](https://github.com/TheAlgorithms/Java/blob/master/src/test/java/com/thealgorithms/dynamicprogramming/LevenshteinDistanceTests.java)
             * [LongestAlternatingSubsequenceTest](https://github.com/TheAlgorithms/Java/blob/master/src/test/java/com/thealgorithms/dynamicprogramming/LongestAlternatingSubsequenceTest.java)
+            * [LongestArithmeticSubsequenceTest](https://github.com/TheAlgorithms/Java/blob/master/src/test/java/com/thealgorithms/dynamicprogramming/LongestArithmeticSubsequenceTest.java)
             * [LongestIncreasingSubsequenceTests](https://github.com/TheAlgorithms/Java/blob/master/src/test/java/com/thealgorithms/dynamicprogramming/LongestIncreasingSubsequenceTests.java)
             * [LongestPalindromicSubstringTest](https://github.com/TheAlgorithms/Java/blob/master/src/test/java/com/thealgorithms/dynamicprogramming/LongestPalindromicSubstringTest.java)
             * [LongestValidParenthesesTest](https://github.com/TheAlgorithms/Java/blob/master/src/test/java/com/thealgorithms/dynamicprogramming/LongestValidParenthesesTest.java)

src/main/java/com/thealgorithms/others/Sudoku.java

@@ -1,33 +1,47 @@
 package com.thealgorithms.others;
 
+/**
+ * A class that provides methods to solve Sudoku puzzles of any n x n size
+ * using a backtracking approach, where n must be a perfect square.
+ * The algorithm checks for safe number placements in rows, columns,
+ * and subgrids (which are sqrt(n) x sqrt(n) in size) and recursively solves the puzzle.
+ * Though commonly used for 9x9 grids, it is adaptable to other valid Sudoku dimensions.
+ */
 final class Sudoku {
+
     private Sudoku() {
     }
 
+    /**
+     * Checks if placing a number in a specific position on the Sudoku board is safe.
+     * The number is considered safe if it does not violate any of the Sudoku rules:
+     * - It should not be present in the same row.
+     * - It should not be present in the same column.
+     * - It should not be present in the corresponding 3x3 subgrid.
+     * - It should not be present in the corresponding subgrid, which is sqrt(n) x sqrt(n) in size (e.g., for a 9x9 grid, the subgrid will be 3x3).
+     *
+     * @param board The current state of the Sudoku board.
+     * @param row   The row index where the number is to be placed.
+     * @param col   The column index where the number is to be placed.
+     * @param num   The number to be placed on the board.
+     * @return True if the placement is safe, otherwise false.
+     */
     public static boolean isSafe(int[][] board, int row, int col, int num) {
-        // Row has the unique (row-clash)
+        // Check the row for duplicates
         for (int d = 0; d < board.length; d++) {
-            // Check if the number we are trying to
-            // place is already present in
-            // that row, return false;
             if (board[row][d] == num) {
                 return false;
             }
         }
 
-        // Column has the unique numbers (column-clash)
+        // Check the column for duplicates
         for (int r = 0; r < board.length; r++) {
-            // Check if the number
-            // we are trying to
-            // place is already present in
-            // that column, return false;
             if (board[r][col] == num) {
                 return false;
             }
         }
 
-        // Corresponding square has
-        // unique number (box-clash)
+        // Check the corresponding 3x3 subgrid for duplicates
         int sqrt = (int) Math.sqrt(board.length);
         int boxRowStart = row - row % sqrt;
         int boxColStart = col - col % sqrt;
@@ -40,22 +54,37 @@ public static boolean isSafe(int[][] board, int row, int col, int num) {
             }
         }
 
-        // if there is no clash, it's safe
         return true;
     }
 
+    /**
+     * Solves the Sudoku puzzle using backtracking.
+     * The algorithm finds an empty cell and tries placing numbers
+     * from 1 to n, where n is the size of the board
+     * (for example, from 1 to 9 in a standard 9x9 Sudoku).
+     * The algorithm finds an empty cell and tries placing numbers from 1 to 9.
+     * The standard version of Sudoku uses numbers from 1 to 9, so the algorithm can be
+     * easily modified for other variations of the game.
+     * If a number placement is valid (checked via `isSafe`), the number is
+     * placed and the function recursively attempts to solve the rest of the puzzle.
+     * If no solution is possible, the number is removed (backtracked),
+     * and the process is repeated.
+     *
+     * @param board The current state of the Sudoku board.
+     * @param n     The size of the Sudoku board (typically 9 for a standard puzzle).
+     * @return True if the Sudoku puzzle is solvable, false otherwise.
+     */
     public static boolean solveSudoku(int[][] board, int n) {
         int row = -1;
         int col = -1;
         boolean isEmpty = true;
+
+        // Find the next empty cell
         for (int i = 0; i < n; i++) {
             for (int j = 0; j < n; j++) {
                 if (board[i][j] == 0) {
                     row = i;
                     col = j;
-
-                    // We still have some remaining
-                    // missing values in Sudoku
                     isEmpty = false;
                     break;
                 }
@@ -70,12 +99,12 @@ public static boolean solveSudoku(int[][] board, int n) {
             return true;
         }
 
-        // Else for each-row backtrack
+        // Try placing numbers 1 to n in the empty cell (n should be a perfect square)
+        // Eg: n=9 for a standard 9x9 Sudoku puzzle, n=16 for a 16x16 puzzle, etc.
         for (int num = 1; num <= n; num++) {
             if (isSafe(board, row, col, num)) {
                 board[row][col] = num;
                 if (solveSudoku(board, n)) {
-                    // print(board, n);
                     return true;
                 } else {
                     // replace it
@@ -86,8 +115,17 @@ public static boolean solveSudoku(int[][] board, int n) {
         return false;
     }
 
+    /**
+     * Prints the current state of the Sudoku board in a readable format.
+     * Each row is printed on a new line, with numbers separated by spaces.
+     *
+     * @param board The current state of the Sudoku board.
+     * @param n     The size of the Sudoku board (typically 9 for a standard puzzle).
+     */
     public static void print(int[][] board, int n) {
-        // We got the answer, just print it
+        // Print the board in a nxn grid format
+        // if n=9, print the board in a 9x9 grid format
+        // if n=16, print the board in a 16x16 grid format
         for (int r = 0; r < n; r++) {
             for (int d = 0; d < n; d++) {
                 System.out.print(board[r][d]);
@@ -101,7 +139,13 @@ public static void print(int[][] board, int n) {
         }
     }
 
-    // Driver Code
+    /**
+     * The driver method to demonstrate solving a Sudoku puzzle.
+     * A sample 9x9 Sudoku puzzle is provided, and the program attempts to solve it
+     * using the `solveSudoku` method. If a solution is found, it is printed to the console.
+     *
+     * @param args Command-line arguments (not used in this program).
+     */
     public static void main(String[] args) {
         int[][] board = new int[][] {
             {3, 0, 6, 5, 0, 8, 4, 0, 0},
@@ -117,7 +161,6 @@ public static void main(String[] args) {
         int n = board.length;
 
         if (solveSudoku(board, n)) {
-            // print solution
             print(board, n);
         } else {
             System.out.println("No solution");