八皇后算法

我们提供一种遍历的思路，我们可以逐行或者逐列来进行可行摆放方案的遍历，每一行（或列）遍历出一个符合条件的位置，接着就到下一行或列遍历下一个棋子的合适位置，这种遍历思路可以保证我们遍历过程中有一个条件是绝对符合的——就是下一个棋子的摆放位置与前面的棋子不在同一行（或列）。接下来，我们只要判断当前位置是否还符合其他条件，如果符合，就遍历下一行（或列）所有位置，看看是否继续有符合条件的位置，以此类推，如果某一个行（或列）的所有位置都不合适，就返回上一行（或列）继续该行（或列）的其他位置遍历，当我们顺利遍历到最后一行（或列），且有符合条件的位置时，就是一个可行的8皇后摆放方案，累加一次八皇后可行方案的个数，然后继续遍历该行其他位置是否有合适的，如果没有，则返回上一行，遍历该行其他位置，依此下去。这样一个过程下来，我们就可以得出所有符合条件的8皇后摆放方案了。这是一个深度优先遍历的过程，同时也是经典的递归思路。

下面用循环实现

public class EightQueensCirculation {    private static final boolean AVAILABLE = true;    private int squares = 8, norm = squares - 1;    private int positionInRow[] = new int[squares];    private int p = -1;    private boolean[] rows = new boolean[squares];    private boolean[] column = new boolean[squares];    private boolean[] leftDiagonal = new boolean[2 * squares - 1];    private boolean[] rightDiagonal = new boolean[2 * squares - 1];    private static int howMany = 0;    public EightQueensCirculation() {        // To complete the initialization work for the        // column,leftDiagonal,rigthDiagonal.        for (int i = 0; i < squares; i++) {            rows[i] = AVAILABLE;            column[i] = AVAILABLE;            positionInRow[i] = -1;        }                for (int i = 0; i < 2 * squares - 1; i++) {            leftDiagonal[i] = AVAILABLE;            rightDiagonal[i] = AVAILABLE;        }    }    public void printResults(int[] columns) {        int row, col;        System.out.println("八皇后问题的第 " + howMany + " 种解法");        System.out.print("八皇后问题的结果为：");        for (int e : columns) {            System.out.print(e);        }        System.out.println("\n具体的图示如下图所示：");        for (row = 0; row < squares; row++) {            for (col = 0; col < squares; col++) {                if (col == positionInRow[row]) {                    System.out.print("@");                } else {                    System.out.print("*");                }            }            System.out.println();        }        System.out.println();    }    public void putQueen() {        int row = 0, col;        while (true) {            for (col = p + 1; col < squares; col++) {                if (rows[row] == AVAILABLE && column[col] == AVAILABLE && leftDiagonal[row + col] == AVAILABLE && rightDiagonal[row - col + norm] == AVAILABLE) {                    break;                }            }            //在当前的行里面找到了可以放置皇后的位置            if (col < squares) {                rows[row] = !AVAILABLE;                column[col] = !AVAILABLE;                leftDiagonal[row + col] = !AVAILABLE;                rightDiagonal[row - col + norm] = !AVAILABLE;                positionInRow[row] = col;                p = col;            } else if (row > 0) {//如果当前行没办反放置皇后了，那么回溯到前一行                row--;                p = positionInRow[row];                rows[row] = AVAILABLE;                column[p] = AVAILABLE;                leftDiagonal[row + p] = AVAILABLE;                rightDiagonal[row - p + norm] = AVAILABLE;                positionInRow[row] = -1;                continue;            } else {                break;            }                        if (row == squares - 1) {                howMany += 1;                printResults(positionInRow);                p = positionInRow[row];                rows[row] = AVAILABLE;                column[p] = AVAILABLE;                leftDiagonal[row + p] = AVAILABLE;                rightDiagonal[row - p + norm] = AVAILABLE;                positionInRow[row] = -1;                continue;            } else {                row++;                p = -1;                continue;            }        }    }    public static void main(String args[]) {        EightQueensCirculation eightQueens = new EightQueensCirculation();        eightQueens.putQueen();        System.out.println("皇后问题一共有" + howMany + "种解法");    }}

下面用递归实现

public class EightQueensRecursive {            private static final boolean AVAILABLE = true;    private int squares = 8, norm = squares-1;    private int positionInRow[] = new int[squares];    private boolean[] column = new boolean[squares];    private boolean[] leftDiagonal = new boolean[2*squares-1];    private boolean[] rightDiagonal = new boolean[2*squares-1];    private static int howMany = 0;    public EightQueensRecursive() {        //To complete the initialization work for the column,leftDiagonal,rigthDiagonal.        for(int i=0; i<squares; i++){            column[i] = AVAILABLE;            positionInRow[i] = -1;        }        for(int i=0; i<2*squares-1; i++){            leftDiagonal[i] = AVAILABLE;            rightDiagonal[i] = AVAILABLE;        }    }    public void printResults(int[] columns) {        int row,col;        System.out.println("八皇后问题的第 "+howMany+" 种解法");        System.out.print("八皇后问题的结果为：");        for(int e:columns) {            System.out.print(e);        }        System.out.println("\n具体的图示如下图所示：");        for(row=0; row<squares; row++) {            for(col=0; col<squares; col++) {                if(col == positionInRow[row]) {                    System.out.print("@");                } else {                    System.out.print("*");                }            }            System.out.println();        }        System.out.println();    }    public void putQueen(int row) {        //如果前面已经得到了一个可行解        for(int i=0; i<squares; i++) {            if(row>squares-1) break;            if(column[i] == AVAILABLE && leftDiagonal[row+i] == AVAILABLE && rightDiagonal[row-i+norm] == AVAILABLE) {                positionInRow[row] = i;                column[i] = !AVAILABLE;                leftDiagonal[row+i] = !AVAILABLE;                rightDiagonal[row-i+norm] = !AVAILABLE;                if(row < squares-1) {                    putQueen(row+1);                } else {                    howMany += 1;                    printResults(positionInRow);                }                column[i] = AVAILABLE;                leftDiagonal[row+i] = AVAILABLE;                rightDiagonal[row-i+norm] = AVAILABLE;            }        }    }    public static void main(String args[]) {        EightQueensRecursive eightQueens = new EightQueensRecursive();        eightQueens.putQueen(0);        System.out.println("皇后问题一共找到了 "+howMany+"组解。");    }}