52. N-Queens II

52. N-Queens II.cpp

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+/*Approach
+NOTE-PLEASE READ APPROACH FIRST THEN SEE THE CODE. YOU WILL DEFINITELY UNDERSTAND THE CODE LINE BY LINE AFTER SEEING THE APPROACH.
+
+Approach for this Problem:
+
+1. Initialize a counter to keep track of the total number of valid N-Queens solutions.
+2. Use three sets to track attacks:
+   - cols: columns where queens are already placed
+   - diag: main diagonals under attack (row + col)
+   - anti_diag: anti-diagonals under attack (row - col)
+3. Create a recursive backtracking function that tries to place a queen row by row.
+4. For each row, iterate through all columns and check if placing a queen at (row, col) is safe (not in cols, diag, or anti_diag).
+5. If safe, mark the column and diagonals as occupied and recurse to the next row.
+6. After recursion, backtrack by unmarking the column and diagonals to explore other possibilities.
+7. When row == n, increment the counter because a valid configuration is found.
+8. Return the counter after exploring all possibilities.
+
+Complexity
+Time complexity: O(n!) - because we try n positions for the first row, n-1 for the next, and so on.
+Space complexity: O(n) - for the recursion stack and sets used to track attacks.
+*/
+
+class Solution {
+public:
+    int totalNQueens(int n) {
+        int count = 0; // step 1: initialize solution counter
+
+        unordered_set<int> cols;      // step 2: columns under attack
+        unordered_set<int> diag;      // step 2: main diagonals under attack
+        unordered_set<int> anti_diag; // step 2: anti-diagonals under attack
+
+        // step 3: recursive backtracking function
+        function<void(int)> backtrack = [&](int row) {
+            if(row == n){ // step 7: valid configuration found
+                count++;
+                return;
+            }
+
+            // step 4: iterate through columns for current row
+            for(int col = 0; col < n; col++){
+                if(cols.count(col) || diag.count(row + col) || anti_diag.count(row - col))
+                    continue; // not safe, skip
+
+                // step 5: place queen and mark attacks
+                cols.insert(col);
+                diag.insert(row + col);
+                anti_diag.insert(row - col);
+
+                backtrack(row + 1); // recurse to next row
+
+                // step 6: backtrack, unmark attacks
+                cols.erase(col);
+                diag.erase(row + col);
+                anti_diag.erase(row - col);
+            }
+        };
+
+        backtrack(0); // start from row 0
+        return count; // step 8: return total solutions
+    }
+};