回溯之N皇后

N皇后问题：在 n x n 的棋盘上放置 N 个皇后，使其不能相互攻击。

1. 问题分析

• 皇后相互攻击是指：在皇后 Q1 的 水平 垂直 正对角线 反对角线 上，都不能有其他皇后。
• 假设：皇后 Q1 的位置在 P1=(x1, y1)、皇后 Q2 的位置在 P2=(x2, y2)，两个皇后相互攻击的问题可以转化为：P1 和 P2 所代表的直线，其斜率不能为：0=水平线、-1=正对角线、1=反对角线、无穷大=垂直线
• 即：k = (y1 == y2) ? NUMBER_MAX : (x1 - x2) / (y1 - y2)，如果 k in (0, 1, -1, NUMBER_MAX)，Q1 和 Q2 能相互攻击，否则 Q1 和 Q2 不能相互攻击。

2. 下面给出 php 版的实现

_showDetail = boolval($showDetail);
    }

    public function nqueen($n){
        $queenPositions = [];
        $res = $this->_backtraceFindNQueenPosition(1, $n, $queenPositions);
        return [$res, $queenPositions, $this->_detailSteps];
    }

    private function _backtraceFindNQueenPosition($row, $totalRows, &$position){
        if($row > $totalRows){
            return true;
        }
        for($col=1; $col<=$totalRows; $col++){
            if($this->_showDetail){
                $tempPosition = $position;
                $tempPosition[$row] = $col;
                array_push($this->_detailSteps, $tempPosition);
            }
            if($this->_areTheseQueenSafe($position, $row, $col)){
                $position[$row] = $col;
                if($this->_backtraceFindNQueenPosition($row+1, $totalRows, $position)){
                    return true;
                }
                unset($position[$row]);
            }
        }
    }

    /**
     * @param $positions
     * @return bool
     */
    private function _areTheseQueenSafe($positions, $row, $col){
        if(0 == count($positions)){
            return true;
        }
        foreach ($positions as $prow => $pcol) {
            $lineK = $row == $prow ? PHP_INT_MAX : ($col - $pcol) / ($row - $prow);
            if(
                $lineK == 0 ||              // same row
                $lineK == PHP_INT_MAX ||    // same col
                $lineK == 1 ||              // on line: left-top to right-bottom (0, 0) is at left-top
                $lineK == -1                // on line: right-top to left-bottom (0, 0) is at left-top
            ){
                return false;
            }
        }
        return true;
    }

    public function printNQueenResult($result, $n){
        list($successed, $position, $steps) = $result;
        if($successed){
            echo 'A solution for '.$n.'-queens is: '.PHP_EOL;
            foreach ($position as $col) {
                echo str_repeat(' x ', $col-1);
                echo ' Q ';
                echo str_repeat(' x ', $n-$col);
                echo PHP_EOL;
            }
            if(0 != count($steps)){
                echo PHP_EOL;
                foreach ($steps as $k => $pos) {
                    echo 'This solution step-'.strval($k+1).' is:'.PHP_EOL;
                    foreach ($pos as $col) {
                        echo str_repeat(' x ', $col-1);
                        echo ' Q ';
                        echo str_repeat(' x ', $n-$col);
                        echo PHP_EOL;
                    }
                }
            }
        }else{
            echo 'No solution found for '.$n.'-queens...'.PHP_EOL;
        }
    }
}
$n = 4;
//$s = new Solution();
$s = new Solution(true);
$res = $s->nqueen($n);
$s->printNQueenResult($res, $n);
echo PHP_EOL;

3. 下面是整个回溯的过程

A solution for 4-queens is: 
 x  Q  x  x 
 x  x  x  Q 
 Q  x  x  x 
 x  x  Q  x 

This solution step-1 is:
 Q  x  x  x 
This solution step-2 is:
 Q  x  x  x 
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This solution step-3 is:
 Q  x  x  x 
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This solution step-4 is:
 Q  x  x  x 
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This solution step-5 is:
 Q  x  x  x 
 x  x  Q  x 
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This solution step-6 is:
 Q  x  x  x 
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This solution step-7 is:
 Q  x  x  x 
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This solution step-8 is:
 Q  x  x  x 
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This solution step-9 is:
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This solution step-10 is:
 Q  x  x  x 
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This solution step-11 is:
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This solution step-12 is:
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This solution step-13 is:
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This solution step-14 is:
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This solution step-15 is:
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This solution step-16 is:
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This solution step-17 is:
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This solution step-18 is:
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This solution step-19 is:
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This solution step-20 is:
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This solution step-21 is:
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This solution step-22 is:
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This solution step-23 is:
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This solution step-24 is:
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This solution step-25 is:
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This solution step-26 is:
 x  Q  x  x 
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 x  x  Q  x 

4. 算法要点

• 回溯的实现是：递归 + 循环 + 回退，即：在 step-n 时，循环选择每一种可能，当 step-[n+x] 证明 step-n 时的选择不正确时，回到 step-n，选择另一种可能。