Fixed: 2554: N Queens Puzzle (#2555)

Co-authored-by: Amit Kumar <akumar@indeed.com>

Backtracking/NQueens.java

@@ -0,0 +1,136 @@
+package Backtracking;
+
+import java.util.ArrayList;
+import java.util.List;
+
+/**
+ * Problem statement:
+ * Given a N x N chess board. Return all arrangements in which N queens can be placed on the board such no two queens attack
+ * each other.
+ * Ex. N = 6
+ * Solution= There are 4 possible ways
+ * Arrangement: 1
+ * ".Q....",
+ * "...Q..",
+ * ".....Q",
+ * "Q.....",
+ * "..Q...",
+ * "....Q."
+ * <p>
+ * Arrangement: 2
+ * "..Q...",
+ * ".....Q",
+ * ".Q....",
+ * "....Q.",
+ * "Q.....",
+ * "...Q.."
+ * <p>
+ * Arrangement: 3
+ * "...Q..",
+ * "Q.....",
+ * "....Q.",
+ * ".Q....",
+ * ".....Q",
+ * "..Q..."
+ * <p>
+ * Arrangement: 4
+ * "....Q.",
+ * "..Q...",
+ * "Q.....",
+ * ".....Q",
+ * "...Q..",
+ * ".Q...."
+ *
+ * Solution:
+ * Brute Force approach:
+ *
+ * Generate all possible arrangement to place N queens on N*N board.
+ * Check each board if queens are placed safely.
+ * If it is safe, include arrangement in solution set. Otherwise ignore it
+ *
+ * Optimized solution:
+ * This can be solved using backtracking in below steps
+ *
+ * Start with first column and place queen on first row
+ * Try placing queen in a row on second column
+ * If placing second queen in second column attacks any of the previous queens, change the row in second column
+ * otherwise move to next column and try to place next queen
+ * In case if there is no rows where a queen can be placed such that it doesn't attack previous queens, then go back to previous column and change row of previous queen.
+ * Keep doing this until last queen is not placed safely.
+ * If there is no such way then return an empty list as solution
+ */
+public class NQueens {
+
+    public static void main(String[] args) {
+        placeQueens(1);
+        placeQueens(2);
+        placeQueens(3);
+        placeQueens(4);
+        placeQueens(5);
+        placeQueens(6);
+    }
+
+    public static void placeQueens(final int queens) {
+        List<List<String>> arrangements = new ArrayList<List<String>>();
+        getSolution(queens, arrangements, new int[queens], 0);
+        if (arrangements.isEmpty()) {
+            System.out.println("There is no way to place " + queens + " queens on board of size " + queens + "x" + queens);
+        } else {
+            System.out.println("Arrangement for placing " + queens + " queens");
+        }
+        arrangements.forEach(arrangement -> {
+            arrangement.forEach(row -> System.out.println(row));
+            System.out.println();
+        });
+    }
+
+    /**
+     * This is backtracking function which tries to place queen recursively
+     * @param boardSize: size of chess board
+     * @param solutions: this holds all possible arrangements
+     * @param columns: columns[i] = rowId where queen is placed in ith column.
+     * @param columnIndex: This is the column in which queen is being placed
+     */
+    private static void getSolution(int boardSize, List<List<String>> solutions, int[] columns, int columnIndex) {
+        if (columnIndex == boardSize) {
+            // this means that all queens have been placed
+            List<String> sol = new ArrayList<String>();
+            for (int i = 0; i < boardSize; i++) {
+                StringBuilder sb = new StringBuilder();
+                for (int j = 0; j < boardSize; j++) {
+                    sb.append(j == columns[i] ? "Q" : ".");
+                }
+                sol.add(sb.toString());
+            }
+            solutions.add(sol);
+            return;
+        }
+
+        // This loop tries to place queen in a row one by one
+        for (int rowIndex = 0; rowIndex < boardSize; rowIndex++) {
+            columns[columnIndex] = rowIndex;
+            if (isPlacedCorrectly(columns, rowIndex, columnIndex)) {
+                // If queen is placed successfully at rowIndex in column=columnIndex then try placing queen in next column
+                getSolution(boardSize, solutions, columns, columnIndex + 1);
+            }
+        }
+    }
+
+    /**
+     * This function checks if queen can be placed at row = rowIndex in column = columnIndex safely
+     * @param columns: columns[i] = rowId where queen is placed in ith column.
+     * @param rowIndex: row in which queen has to be placed
+     * @param columnIndex: column in which queen is being placed
+     * @return true: if queen can be placed safely
+     * false: otherwise
+     */
+    private static boolean isPlacedCorrectly(int[] columns, int rowIndex, int columnIndex) {
+        for (int i = 0; i < columnIndex; i++) {
+            int diff = Math.abs(columns[i] - rowIndex);
+            if (diff == 0 || columnIndex - i == diff) {
+                return false;
+            }
+        }
+        return true;
+    }
+}