本文中的代码在文章最后提供下载链接。

关于八皇后的问题,自己百度。回溯法的概念也可以自己维基百科中查看。

回溯法无非就是遍历解空间,如果合适的话就继续,不合适的话就放弃当前节点。可以采用递归实现。八皇后问题的经典算法就是采用回溯法。网上也有许多版本的解法,但是C#版本的似乎很少。笔者在此奉献上。

此处八皇后的算法的伪代码如下:

在第row行放置皇后(row)

{

for 第row行的格子

{           

把当前位置加入作为候选解加入解决方案中

   If 目前解决方案没有问题

           {

            再下一行放皇后,递归调用PlaceQueen(row+1);

           如果是最后一行放完了,则成功,打印棋盘

            }

    else

                          将位置(row,col)上的皇后取走,尝试下一列位置  //回溯

    }

伪代码的算法能理解之后,就可以写代码了。

对于棋盘,可以使用二维数组实现,但是本例中使用的是一个List序列,序列中有64个格子的信息。每个格子的定义如下。

//格子类。坐标x,y

   

  1. public class position  
  2.    {  
  3.        public int x { setget; }  
  4.        public int y { setget; }  
  5.        public override string ToString()  
  6.        {  
  7.            return string.Format("[{0},{1}]", x, y);  
  8.        }  
  9.    }  

 

初始化格子的代码如下:

 

  1. #region 初始化格子  
  2.             for (int i = 1; i <= dimension; i++)  
  3.             {  
  4.                 for (int j = 1; j <= dimension; j++)  
  5.                 {  
  6.                     lstAllPosition.Add(new position { x = i, y = j });  
  7.                 }  
  8.             }  
  9.             #endregion 

 

这样的话lstAllPosition就存放了64个格子了。由于采用了C#作为实现的语言,因此尽情的使用他的一些优点,比如采用Linq和拉姆达表达式来选定特定的行的元素。此处,需要有C#的一些知识。

对于放置皇后的实现。本人写了2种方法,2中方法本质都是一样的,只不过理解起来稍微有些不同。

第一种方法中规中矩。

看如下代码和注释: 

  1. static void PlaceQueen(int row)  
  2.         {  
  3.             #region 实现方法一  
  4.             //当前行的元素  
  5.             var currentrow = lstAllPosition.Where(m => m.x == row).ToList();  
  6.             for (int index = 0; index < currentrow.Count;index++ )  
  7.             {  
  8.                 //加入一个元素  
  9.                 solution.Add(currentrow[index]);  
  10.                 //如果有效的话,继续下面的递归  
  11.                 if (Valid(solution))  
  12.                 {  
  13.                     //如果row是最后一行了,说明已经满足。打印。  
  14.                     if (row == dimension)  
  15.                     {  
  16.                         print(solution);  
  17.                         //找到一组解之后,删除该元素,继续下一组解的查找  
  18.                         solution.Remove(solution.Last());  
  19.                     }  
  20.                     else 
  21.                     {  
  22.                         //递归调用下一行  
  23.                         PlaceQueen(row + 1);  
  24.                     }  
  25.                 }  
  26.                 else 
  27.                 {  
  28.                     solution.Remove(currentrow[index]);  
  29.                 }  
  30.                 //如果该行的元素每个都被试过了,但是还是没能运行到print(solution)方法中去,则该方法即将运行结束,结束前,把最后一个元素也删掉。  
  31.                 //由于该方法是递归的,因此结束之后,会继续重新下一个元素。  
  32.                 if (index == dimension-1 && solution.Count > 0)  
  33.                 {  
  34.                     solution.Remove(solution.Last());  
  35.                 }  
  36.             }  
  37.             #endregion  
  38.         }  
  39.  

 

            第二种方法也很好理解,只是把solution.Remove(item);删除元素的语句写到外面,表示,无论怎样,都会把刚加入的元素删除。

但是在删除元素之前,由于递归调用的结果,在某些情况下solution列的元素加满到8个时候,还没删除元素。这是就直接打印出结果,然后再删除最后一个元素。很容易推断,到最后一个元素放入solution的时候,肯定是递归嵌套了8层了。  

  1. static void PlaceQueen(int row)  
  2.         {  
  3.             #region 实现方法二  
  4.             foreach (var item in lstAllPosition.Where(m =&gt; m.x == row).ToList())  
  5.             {  
  6.                 //Console.WriteLine("-------");  
  7.                 solution.Add(item);  
  8.                 //solution.ForEach(m =&gt; Console.Write(m.ToString()));  
  9.                 //Console.WriteLine();  
  10.                 //如果有效的话,继续下面的递归  
  11.                 if (Valid(solution))  
  12.                 {  
  13.                     //如果row是最后一行了,说明已经满足。打印。  
  14.                     if (row == dimension)  
  15.                     {  
  16.                         print(solution);  
  17.                     }  
  18.                     else 
  19.                     {  
  20.                         //递归调用下一行  
  21.                         PlaceQueen(row + 1);  
  22.                     }  
  23.                 }  
  24.                 //Console.WriteLine("***");  
  25.                 //把当前元素去除,尝试下一个元素。此语句,每一轮都会执行,除非该轮的调用,  
  26.                 //每次递归调用PlaceQueen时Valid(solution)都是true,也就是说找到符合条件的结果集,打印完后。  
  27.                 //还是会继续删除最厚放入的元素,继续尝试下一个元素是否匹配。  
  28.                 solution.Remove(item);  
  29.               //solution.ForEach(m =&gt; Console.Write(m.ToString()));  
  30.                 //Console.WriteLine();  
  31.                 //Console.WriteLine("-------");  
  32.             }  
  33.             #endregion  
  34.         }  
  35.  

其中 验证当前集合是否有效的函数如下:

  //验证有效性

       

  1. static bool Valid(List<position> solve)  
  2.        {  
  3.            var item2 = solve.Last();  
  4.            foreach (var item1 in solve)//把最后加入的元素后之前每一个元素做比较。  
  5.            {  
  6.                if (item1 == item2)  
  7.                {  
  8.                    break;  
  9.                }  
  10.                if (item1.y == item2.y)//y轴相同  
  11.                {  
  12.                    return false;  
  13.                }  
  14.                else if (Math.Abs(decimal.Parse((item1.y - item2.y).ToString()) / (item1.x - item2.x)) == 1)//斜线  
  15.                {  
  16.                    return false;  
  17.                }  
  18.            }  
  19.            return true;  
  20.        } 

打印棋盘的方法如下,使用了Linq和拉姆达表达式。

//打印棋盘

       

  1. static void print(List<position> solve)  
  2.        {  
  3.            Console.ForegroundColor = ConsoleColor.Gray;  
  4.            Console.WriteLine("-----" + (count++) + "-------");  
  5.            for (int i = 1; i <= dimension; i++)  
  6.            {  
  7.                foreach (var item in lstAllPosition.Where(m => m.x == i).ToList())  
  8.                {  
  9.                    if (solve.Contains(item))  
  10.                    {  
  11.                        Console.ForegroundColor = ConsoleColor.Yellow;  
  12.                        Console.Write("1 ");  
  13.                    }  
  14.                    else 
  15.                    {  
  16.                        Console.ForegroundColor = ConsoleColor.Gray;  
  17.                        Console.Write("0 ");  
  18.                    }  
  19.                }  
  20.                Console.WriteLine();  
  21.            }  
  22.        } 

 结果如下:共92个解。

image

所有的解集如下:

-----1-------
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
-----2-------
1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
-----3-------
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0
-----4-------
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
-----5-------
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
-----6-------
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0
-----7-------
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
-----8-------
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0
-----9-------
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
-----10-------
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
-----11-------
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
-----12-------
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
-----13-------
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
-----14-------
0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
-----15-------
0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
-----16-------
0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
-----17-------
0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
-----18-------
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
-----19-------
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
-----20-------
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 1 0 0 0 0
-----21-------
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
-----22-------
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
-----23-------
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
-----24-------
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
-----25-------
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
-----26-------
0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
-----27-------
0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
-----28-------
0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
-----29-------
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
-----30-------
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
-----31-------
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0
-----32-------
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
-----33-------
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
-----34-------
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
-----35-------
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
-----36-------
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
-----37-------
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0
-----38-------
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0
-----39-------
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
-----40-------
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
-----41-------
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0
-----42-------
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1
-----43-------
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
-----44-------
0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
-----45-------
0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
-----46-------
0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
-----47-------
0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0
-----48-------
0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
-----49-------
0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
-----50-------
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
-----51-------
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0
-----52-------
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0
-----53-------
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
-----54-------
0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
-----55-------
0 0 0 0 1 0 0 0
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