算法回顾--N皇后问题简单回顾

前言

最近学习的过程中，不知道哪门子的思维发散，突然又遇见皇后问题了，于是乎老调重弹，心里琢磨，虽然思路大家都容易懂，哪怕是最简单的野蛮回溯法，说着简单，但是如果非得编码实现？我可以一次性写出来OK的代码吗？我对此表示疑问，于是乎动手写代码，发现写此类算法问题，最重要的是边界条件的判断。这里说明一下，这篇纯属练手，不考虑算法效率，只是为了实现，为了练习最暴力野蛮的回溯，说白了，就是怎样简单粗暴的玩弄for和while这些个玩意！

实现

本人比较懒，所以懒得搞二维数组来存储皇后坐标，其实用二维数组来存储，然后根据行列来遍历和匹配，确实效率高了不少，但是，我在这里仅仅只是练习回溯的思想，所以只是使用简单的数组来存储N*N个数，比如8皇后问题，我就用数组来存储 {1,2,3,,,,,,,,,,64}，然后使用queens数组存储安全的皇后

代码

/// <summary>
/// 皇后算法非递归实现，主要思想如下：
/// 初始化皇后备选列表为空（安全），将所有坐标的棋子尝试放入
/// 如果符合安全，则放入（如果数目达到则保留输出），如果不安全则跳过继续搜索
/// 如果搜索到结尾依然未搜到，则循环向上回溯到上一次解的下一个继续
/// </summary>
public class Queen
{
    // 矩阵维度
    int size;
    // 从1编号，存放所有整数
    int[] array;
    int count;

    public Queen()
        : this(4)
    {
    }

    public Queen(int s)
    {
        this.size = s;
        InitList();
    }

    private void InitList()
    {
        count = size * size;
        array = new int[count];
        for (int i = 0; i < count; i++)
        {
            array[i] = i + 1;
        }
    }

    public void PrintQueenResult()
    {
        List<int> queens = new List<int>();

        // 记录探测索引
        int tryIndex = 0;
        // 记录最后一次皇后解
        int lastQueen = 0;
        // 记录皇后总数
        int queenCount = 0;
        // 记录所有全皇后解个数
        int solutions = 0;
        while (true)
        {
            // 探测元素
            int tryItem = array[tryIndex];

            // 如果探测到最后一个元素，并且皇后已经回溯为空
            // 表示回溯已经全部完成，返回
            if (tryItem == array[count - 1] && queenCount == 0)
                break;

            // 找到安全的皇后
            if (IsSafe(queens, tryItem))
            {
                queens.Add(tryItem);
                // 皇后数加1
                queenCount++;
                // 记录最后一次放置的皇后
                lastQueen = tryItem;

                // 如果皇后获取足够，输出
                if (queenCount == size)
                {
                    Console.WriteLine("找到皇后排列：{0}",
                        string.Join(",", queens));
                    solutions++;
                }
            }
            else
            {
                if (tryIndex < count - 1)
                {
                    // 如果不安全，还未搜索结束，继续搜索
                    tryIndex++;
                }
                else
                {
                    // 如果不安全，但搜索结束，则回溯
                    while (tryIndex >= count - 1)
                    {
                        queens.Remove(lastQueen);
                        queenCount--;
                        tryIndex = lastQueen;

                        // 如果回溯完毕则跳出
                        if (queenCount == 0)
                            break;

                        lastQueen = queens[queenCount - 1];
                    }
                }
            }
        }

        Console.WriteLine("全部解个数为：{0}", solutions);
    }

    private bool IsSafe(List<int> list, int target)
    {
        if (list == null || list.Count == 0)
            return true;
        return list.All(d => !IsDangerous(d, target));
    }

    private bool IsDangerous(int a, int b)
    {
        int rowA = (a - 1) / size + 1;
        int rowB = (b - 1) / size + 1;
        int colA = (a - 1) % size + 1;
        int colB = (b - 1) % size + 1;

        if (a == b) return true;
        if (rowA == rowB) return true;
        if (colA == colB) return true;
        if (Math.Abs(rowA - rowB) == Math.Abs(colA - colB))
            return true;

        return false;
    }

    private int Pop(List<int> list)
    {
        int last = list[list.Count - 1];
        list.Remove(last);
        return last;
    }

    private int Peek(List<int> list)
    {
        return list[list.Count - 1];
    }
}

测试

八皇后测试代码：Queen q = new Queen(8);  q.PrintQueenResult();

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" alt="" />

结语

脑子不练会生锈，还是要经常做做这些最基础的东西，把不习惯变成习惯，慢慢来，不早了，先睡了