C#，动态规划（DP）N皇后问题（N Queen Problem）的回溯（Backtracking）算法与源代码-编程知识

1 N皇后问题（N Queen Problem）

在N*N的方格棋盘放置了N个皇后，使得它们不相互攻击（即任意2个皇后不允许处在同一排，同一列，也不允许处在与棋盘边框成45角的斜线上。

2 回溯算法

回溯算法实际上一个类似枚举的搜索尝试过程，主要是在搜索尝试过程中寻找问题的解，当发现已不满足求解条件时，就“回溯”返回，尝试别的路径。回溯法是一种选优搜索法，按选优条件向前搜索，以达到目标。但当探索到某一步时，发现原先选择并不优或达不到目标，就退回一步重新选择，这种走不通就退回再走的技术为回溯法，而满足回溯条件的某个状态的点称为“回溯点”。

回溯法也称试探法，它的基本思想是：从问题的某一种状态（初始状态）出发，搜索从这种状态出发所能达到的所有“状态”，当一条路走到“尽头”的时候（不能再前进），再后退一步或若干步，从另一种可能“状态”出发，继续搜索，直到所有的“路径”（状态）都试探过。这种不断“前进”、不断“回溯”寻找解的方法，就称作“回溯法”。

3 源程序

using System;
using System.Text;
using System.Collections;
using System.Collections.Generic;

namespace Legalsoft.Truffer.Algorithm
{
   /// <summary>
   /// N皇后问题
   /// </summary>
   public static partial class Algorithm_Gallery
   {
   private static bool NQP_IsSafe(int[,] board, int row, int col)
       {
           int N = board.GetLength(0);
           for (int i = 0; i < col; i++)
           {
               if (board[row, i] == 1)
               {
                   return false;
               }
           }
           for (int i = row, j = col; i >= 0 && j >= 0; i--, j--)
           {
               if (board[i, j] == 1)
               {
                   return false;
               }
           }
           for (int i = row, j = col; j >= 0 && i < N; i++, j--)
           {
               if (board[i, j] == 1)
               {
                   return false;
               }
           }
           return true;
       }

       private static bool NQP_Utility(ref int[,] board, int col)
       {
           int N = board.GetLength(0);
           if (col >= N)
           {
               return true;
           }
           for (int i = 0; i < N; i++)
           {
               if (NQP_IsSafe(board, i, col))
               {
                   board[i, col] = 1;

                   if (NQP_Utility(ref board, col + 1) == true)
                   {
                       return true;
                   }
                   board[i, col] = 0;
               }
           }
           return false;
       }

       public static bool NQP_Solve(int n,out int[,] board)
       {
           board = new int[n, n];

           if (NQP_Utility(ref board, 0) == false)
           {
               return false;
           }

return true;
}

       public static string ToHtml(int[,] board)
       {
           int N = board.GetLength(0);
           StringBuilder sb = new StringBuilder();
           sb.AppendLine("<style>");
           sb.AppendLine("td { padding:5px;text-align:center; }");
           sb.AppendLine("</style>");
           sb.AppendLine("<table border=1 bordercolor='#999999' style='border-collapse:collapse;'>");
           for (int i = 0; i < N; i++)
           {
               sb.AppendLine("<tr>");
               for (int j = 0; j < N; j++)
               {
                   sb.AppendLine("<td>" + board[i, j] + "</td>");
               }
               sb.AppendLine("</tr>");
           }
           sb.AppendLine("</table>");
           return sb.ToString();
       }
   }
}

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4 源代码

using System;
using System.Text;
using System.Collections;
using System.Collections.Generic;namespace Legalsoft.Truffer.Algorithm
{/// <summary>/// N皇后问题/// </summary>public static partial class Algorithm_Gallery{private static bool NQP_IsSafe(int[,] board, int row, int col){int N = board.GetLength(0);for (int i = 0; i < col; i++){if (board[row, i] == 1){return false;}}for (int i = row, j = col; i >= 0 && j >= 0; i--, j--){if (board[i, j] == 1){return false;}}for (int i = row, j = col; j >= 0 && i < N; i++, j--){if (board[i, j] == 1){return false;}}return true;}private static bool NQP_Utility(ref int[,] board, int col){int N = board.GetLength(0);if (col >= N){return true;}for (int i = 0; i < N; i++){if (NQP_IsSafe(board, i, col)){board[i, col] = 1;if (NQP_Utility(ref board, col + 1) == true){return true;}board[i, col] = 0;}}return false;}public static bool NQP_Solve(int n,out int[,] board){board = new int[n, n];            if (NQP_Utility(ref board, 0) == false){return false;}return true;}public static string ToHtml(int[,] board){int N = board.GetLength(0);StringBuilder sb = new StringBuilder();sb.AppendLine("<style>");sb.AppendLine("td { padding:5px;text-align:center; }");sb.AppendLine("</style>");sb.AppendLine("<table border=1 bordercolor='#999999' style='border-collapse:collapse;'>");for (int i = 0; i < N; i++){sb.AppendLine("<tr>");for (int j = 0; j < N; j++){sb.AppendLine("<td>" + board[i, j] + "</td>");}sb.AppendLine("</tr>");}sb.AppendLine("</table>");return sb.ToString();}}
}