Given the dimension of a chess board (N x M), determine the minimum number of queens required to cover all the squares of the board. A queen can attack any square along its row, column or diagonals.
Examples:
Input : N = 8, M = 8
Output : 5
Layout : Q X X X X X X X
X X Q X X X X X
X X X X Q X X X
X Q X X X X X X
X X X Q X X X X
X X X X X X X X
X X X X X X X X
X X X X X X X X
Input : N = 3, M = 5
Output : 2
Layout : Q X X X X
X X X X X
X X X Q X
This article attempts to solve the problem in a very simple way without much optimization.
Step 1: Starting from any corner square of the board, find an ‘uncovered’ square (Uncovered square is a square which isn’t attacked by any of the queens already placed). If none found, goto Step 4.
Step 2: Place a Queen on this square and increment variable ‘count’ by 1.
Step 3: Repeat step 1.
Step 4: Now, you’ve got a layout where every square is covered. Therefore, the value of ‘count’ can be the answer. However, you might be able to do better, as there might exist a better layout with lesser number of queens. So, store this ‘count’ as the best value till now and proceed to find a better solution.
Step 5: Remove the last queen placed and place it in the next ‘uncovered’ cell.
Step 6: Proceed recursively and try out all the possible layouts. Finally, the one with the least number of queens is the answer.
Dry run the following code for better understanding.
// Java program to find minimum number of queens needed // to cover a given chess board. public class Backtracking { // The chessboard is represented by a 2D array. static boolean[][] board; // N x M is the dimension of the chess board. static int N, M; // The minimum number of queens required. // Initially, set to MAX_VAL. static int minCount = Integer.MAX_VALUE; static String layout; // Stores the best layout. // Driver code public static void main(String[] args) { N = 8; M = 8; board = new boolean[N][M]; placeQueen(0); System.out.println(minCount); System.out.println("\nLayout: \n" + layout); } // Finds minimum count of queens needed and places them. static void placeQueen(int countSoFar) { int i, j; if (countSoFar >= minCount) // We've already obtained a solution with lesser or // same number of queens. Hence, no need to proceed. return; // Checks if there exists any unattacked cells. findUnattackedCell : { for (i = 0; i < N; ++i) for (j = 0; j < M; ++j) if (!isAttacked(i, j)) // Square (i, j) is unattacked. break findUnattackedCell; // All squares all covered. Hence, this // is the best solution till now. minCount = countSoFar; storeLayout(); return; } for (i = 0; i < N; ++i) for (j = 0; j < M; ++j) { if (!isAttacked(i, j)) { // Square (i, j) is unattacked. // Therefore, place a queen here. board[i][j] = true; // Increment 'count' and proceed recursively. placeQueen(countSoFar + 1); // Remove this queen and attempt to // find a better solution. board[i][j] = false; } } } // Returns 'true' if the square (row, col) is // being attacked by at least one queen. static boolean isAttacked(int row, int col) { int i, j; // Check the 'col'th column for any queen. for (i = 0; i < N; ++i) if (board[i][col]) return true; // Check the 'row'th row for any queen. for (j = 0; j < M; ++j) if (board[row][j]) return true; // Check the diagonals for any queen. for (i = 0; i < Math.min(N, M); ++i) if (row - i >= 0 && col - i >= 0 && board[row - i][col - i]) return true; else if (row - i >= 0 && col + i < M && board[row - i][col + i]) return true; else if (row + i < N && col - i >= 0 && board[row + i][col - i]) return true; else if (row + i < N && col + i < M && board[row + i][col + i]) return true; // This square is unattacked. Hence return 'false'. return false; } // Stores the current layout in 'layout' // variable as String. static void storeLayout() { StringBuilder sb = new StringBuilder(); for (boolean[] row : board) { for (boolean cell : row) sb.append(cell ? "Q " : "X "); sb.append("\n"); } layout = sb.toString(); } } |
5 Layout: Q X X X X X X X X X Q X X X X X X X X X Q X X X X Q X X X X X X X X X Q X X X X X X X X X X X X X X X X X X X X X X X X X X X X
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