迷宫问题
今天看到个很经典的迷宫问题，这个太经典了 非常适合研究算法方面。
迷宫无非就是一个地图然后找出走出迷宫的线路，经典的递归问题。
废话不多说直接看吧！
1.经典迷宫
2.复杂迷宫问题
3.最优路线迷宫问题
1.经典迷宫
先看地图
从图上标出的起点到终点找出线路的算法
先看地图，地图无非就是一个二维的数组，用某个标记标记为墙壁（这里用数字9）某个标记标记成线路（这里用数字1）其他直接是空白
看看上面迷宫就知道这个二维数组是什么样的了吧
class Program { static int[][] map = new int[9][]; static void Main(string[] args) { map[0] = new int[] { 9, 9, 9, 9, 9, 9, 9, 9, 9 }; map[1] = new int[] { 9, 0, 0, 0, 0, 0, 0, 0, 9 }; map[2] = new int[] { 9, 0, 9, 9, 0, 9, 9, 0, 9 }; map[3] = new int[] { 9, 0, 9, 0, 0, 9, 0, 0, 9 }; map[4] = new int[] { 9, 0, 9, 0, 9, 0, 9, 0, 9 }; map[5] = new int[] { 9, 0, 0, 0, 0, 0, 9, 0, 9 }; map[6] = new int[] { 9, 9, 0, 9, 9, 0, 9, 9, 9 }; map[7] = new int[] { 9, 0, 0, 0, 0, 0, 0, 0, 9 }; map[8] = new int[] { 9, 9, 9, 9, 9, 9, 9, 9, 9 }; Console.WriteLine("迷宫地图："); for (int i = 0; i < map.Length; i++) { for (int j = 0; j < map.Length; j++) if (map[i][j] == 9) Console.Write("██"); else Console.Write(" "); Console.WriteLine(""); } System.Console.ReadKey(); } }
然后确定起始点和结束点就可以用递归去计算并输出线路了：
1 static int startX = 1, startY = 1; 2 static int endX = 7, endY = 7; 3 4 static void Go(int i, int j) { 5 int m, n; 6 map[i][j] = 1; 7 if (i == endY && j == endX) { 8 Console.WriteLine("\n迷宫行走路线"); 9 for (m = 0; m < map.Length; m++) { 10 for (n = 0; n < map.Length; n++) 11 if (map[m][n] == 9) 12 Console.Write("██"); 13 else if (map[m][n] == 1) { 14 Console.Write("◇"); 15 } 16 else 17 Console.Write(" "); 18 Console.WriteLine(""); 19 } 20 } 21 if (map[i][j + 1] == 0) 22 Go(i, j + 1); 23 if (map[i + 1][j] == 0) 24 Go(i + 1, j); 25 if (map[i][j - 1] == 0) 26 Go(i, j - 1); 27 if (map[i - 1][j] == 0) 28 Go(i - 1, j); 29 map[i][j] = 0; 30 }
当到达结束点的时候就输出线路，这个应该好理解，后面的逐个方向去递归求解看看也就会了。
2.复杂迷宫问题
先看地图
然后看解法运行结果：
这是一个20*20的迷宫跟上面一样用9标记成墙壁
先打印地图：
class Program { static int[][] map; static void Main(string[] args) { map = new int[20][] { new int[] { 9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9}, new int[] { 9,0,0,0,9,0,0,9,9,9,9,9,9,9,9,9,9,9,9,9}, new int[] { 9,0,9,0,0,0,0,9,9,9,9,9,9,9,9,9,9,9,9,9}, new int[] { 9,0,9,9,9,9,0,0,0,0,0,9,9,9,9,9,9,9,9,9}, new int[] { 9,0,9,9,0,0,0,9,0,9,0,9,9,9,9,9,9,9,9,9}, new int[] { 9,0,0,0,0,9,9,9,0,9,0,0,0,0,9,9,9,9,9,9}, new int[] { 9,9,0,9,9,9,9,9,0,9,9,9,9,0,0,0,9,9,9,9}, new int[] { 9,9,0,9,9,9,9,9,9,9,9,9,9,0,9,9,9,9,9,9}, new int[] { 9,9,0,9,9,9,9,9,9,9,0,0,0,0,9,9,9,9,9,9}, new int[] { 9,0,0,9,9,9,9,9,9,9,0,9,9,9,9,9,9,9,9,9}, new int[] { 9,0,9,9,9,9,9,9,9,9,0,9,9,9,0,9,9,9,9,9}, new int[] { 9,0,9,9,9,9,9,9,9,9,0,0,0,0,9,9,9,9,9,9}, new int[] { 9,0,9,9,9,9,9,9,9,9,9,9,0,9,9,9,9,9,9,9}, new int[] { 9,0,0,9,9,9,9,9,9,9,9,9,0,9,9,9,9,9,9,9}, new int[] { 9,9,0,9,9,0,0,0,0,0,0,0,0,9,9,9,9,9,9,9}, new int[] { 9,9,0,9,9,0,9,9,9,9,9,9,0,9,9,9,9,9,9,9}, new int[] { 9,9,0,9,9,0,9,9,9,9,9,9,0,9,9,9,9,9,9,9}, new int[] { 9,9,0,9,9,0,9,9,9,9,9,9,0,9,9,9,9,9,9,9}, new int[] { 9,9,0,0,0,0,0,0,0,9,9,9,0,0,0,0,0,0,0,9}, new int[] { 9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9} }; Console.ForegroundColor = ConsoleColor.White; Console.WriteLine("迷宫地图："); Console.ForegroundColor = ConsoleColor.Green; for (int i = 0; i < map.Length; i++) { for (int j = 0; j < map.Length; j++) { if (map[i][j] == 9) Console.Write("■"); else Console.Write(" "); } Console.WriteLine(" "); } System.Console.ReadKey(); } }
这里能走的线路比较多所以每条正确线路都用一个结果集进行保存，方便后面求出最短线路
static int startX = 1, startY = 1;//起点 static int endX = 18, endY = 18;//终点 static List<List<string>> list = new List<List<string>>(); static List<string> lis = new List<string>(); static void go(int i, int j) { int m, n; map[i][j] = 1; if (i == endY && j == endX) { Console.ForegroundColor = ConsoleColor.White; Console.WriteLine("\n迷宫行走路线{0}", list.Count); lis.Clear(); for (m = 0; m < map.Length; m++) { for (n = 0; n < map.Length; n++) if (map[m][n] == 9) { Console.ForegroundColor = ConsoleColor.DarkRed; Console.Write("■"); } else if (map[m][n] == 1) { Console.ForegroundColor = ConsoleColor.Yellow; lis.Add("" + m + "," + n); Console.BackgroundColor = ConsoleColor.Blue; Console.Write("{0}", lis.Count.ToString().PadLeft(2, ' ')); Console.BackgroundColor = ConsoleColor.Black; } else { Console.ForegroundColor = ConsoleColor.Yellow; Console.Write("->"); Console.ForegroundColor = ConsoleColor.Black; } Console.WriteLine(""); } list.Add((lis.ToArray().Clone() as string[]).ToList<string>()); Console.ReadKey(); } if (map[i][j + 1] == 0) go(i, j + 1); //下 if (map[i + 1][j] == 0) //左 go(i + 1, j); if (map[i][j - 1] == 0) //上 go(i, j - 1); if (map[i - 1][j] == 0) //右 go(i - 1, j); map[i][j] = 0; }
走完之后直接在结果集中就可以找出最优线路了：
Console.ForegroundColor = ConsoleColor.White; Console.WriteLine("\n最优路线："); list.OrderBy(a => a.Count).First().ForEach(b =>Console.Write("["+b.ToString() +"]"+ " ")); System.Console.ReadKey();
3.最优路线迷宫问题
前面两种迷宫都还是比较简单的，一看就明白了。下面看看更大更复杂的迷宫进行最优线路选择问题：
一看就头好晕啊，这么烦的东西肯定是用EXCEL这些工具帮忙画出来的然后结合成代码复制黏贴一下的啦。
1 class Program { 2 static int[][] map; 3 static void Main(string[] args) { 4 map = new int[40][] { 5 new int[] { 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, }, 6 new int[] { 9, 0, 9, 0, 0, 0, 9, 0, 0, 0, 0, 0, 9, 9, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 9, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 9, }, 7 new int[] { 9, 0, 9, 0, 9, 0, 9, 0, 9, 9, 9, 0, 9, 9, 0, 9, 9, 9, 9, 0, 9, 0, 9, 9, 0, 9, 0, 9, 9, 0, 9, 0, 9, 9, 9, 9, 0, 9, 9, 9, }, 8 new int[] { 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 0, 9, 0, 9, 9, 0, 9, 0, 9, 0, 9, 9, 0, 9, 0, 9, 9, 0, 9, 0, 9, 9, 9, 9, 0, 9, 9, 9, }, 9 new int[] { 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 9, 0, 9, 0, 0, 0, 0, 9, 0, 9, 9, 0, 9, 0, 0, 0, 0, 9, 0, 9, 9, 9, 9, 0, 9, 9, 9, }, 10 new int[] { 9, 0, 9, 0, 9, 0, 9, 0, 0, 0, 9, 0, 9, 9, 0, 9, 9, 0, 9, 9, 9, 0, 0, 9, 0, 9, 0, 9, 9, 0, 9, 0, 0, 0, 0, 0, 0, 9, 9, 9, }, 11 new int[] { 9, 0, 9, 0, 9, 0, 9, 9, 9, 0, 9, 0, 9, 0, 0, 9, 9, 0, 9, 9, 9, 9, 0, 9, 0, 0, 0, 9, 9, 0, 9, 0, 9, 9, 9, 9, 0, 9, 9, 9, }, 12 new int[] { 9, 0, 9, 0, 9, 0, 9, 9, 9, 0, 9, 0, 9, 9, 0, 9, 0, 0, 0, 0, 0, 9, 0, 9, 9, 9, 9, 9, 9, 0, 9, 0, 9, 9, 9, 9, 0, 9, 9, 9, }, 13 new int[] { 9, 0, 9, 0, 9, 0, 0, 0, 0, 0, 9, 0, 9, 9, 0, 9, 0, 9, 9, 9, 0, 9, 0, 9, 9, 9, 9, 9, 9, 0, 9, 0, 9, 9, 9, 9, 0, 9, 9, 9, }, 14 new int[] { 9, 0, 9, 0, 9, 9, 0, 9, 9, 9, 9, 0, 9, 9, 0, 9, 0, 9, 9, 9, 0, 9, 0, 9, 9, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 9, }, 15 new int[] { 9, 0, 9, 0, 9, 9, 0, 9, 9, 0, 9, 0, 9, 9, 0, 9, 0, 9, 0, 0, 0, 9, 0, 9, 9, 0, 9, 9, 9, 9, 9, 0, 9, 0, 9, 9, 9, 9, 0, 9, }, 16 new int[] { 9, 0, 9, 0, 9, 9, 0, 0, 9, 0, 0, 0, 9, 9, 0, 9, 0, 9, 0, 9, 9, 9, 0, 0, 9, 0, 9, 0, 0, 0, 0, 0, 9, 0, 9, 0, 0, 0, 0, 9, }, 17 new int[] { 9, 0, 9, 0, 9, 9, 9, 0, 9, 0, 9, 9, 9, 9, 0, 9, 9, 9, 0, 9, 9, 9, 9, 0, 9, 0, 9, 0, 9, 9, 9, 9, 9, 0, 9, 0, 9, 9, 0, 9, }, 18 new int[] { 9, 0, 9, 0, 9, 9, 9, 0, 9, 0, 9, 9, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 9, 0, 0, 0, 9, 0, 9, 0, 0, 0, 0, 0, 9, 0, 9, 9, 0, 9, }, 19 new int[] { 9, 0, 9, 0, 9, 9, 9, 9, 9, 0, 9, 9, 0, 9, 9, 9, 0, 9, 9, 9, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 9, 9, 9, 9, 0, 9, 9, 0, 9, }, 20 new int[] { 9, 0, 9, 0, 0, 0, 0, 0, 9, 0, 9, 9, 0, 9, 9, 9, 0, 0, 9, 9, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 0, 0, 0, 0, 9, 0, 0, 9, }, 21 new int[] { 9, 0, 9, 9, 9, 0, 9, 9, 9, 0, 9, 9, 0, 0, 0, 9, 9, 0, 9, 0, 0, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 9, 9, 9, 9, 0, 9, 9, }, 22 new int[] { 9, 0, 9, 9, 9, 0, 9, 9, 9, 0, 0, 0, 0, 9, 0, 9, 9, 0, 9, 0, 9, 9, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 9, 9, 9, 9, 0, 9, 9, }, 23 new int[] { 9, 0, 0, 9, 9, 0, 0, 0, 9, 0, 9, 9, 0, 9, 0, 9, 9, 0, 9, 0, 9, 9, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 9, 9, 9, 9, 0, 9, 9, }, 24 new int[] { 9, 9, 0, 0, 9, 0, 9, 0, 9, 0, 9, 9, 9, 9, 0, 9, 9, 0, 9, 0, 0, 0, 0, 0, 9, 0, 0, 0, 9, 0, 9, 0, 0, 0, 0, 0, 0, 0, 9, 9, }, 25 new int[] { 9, 9, 9, 0, 9, 9, 9, 0, 9, 0, 9, 9, 9, 9, 0, 9, 9, 0, 9, 9, 0, 9, 9, 9, 9, 9, 9, 0, 9, 0, 9, 0, 9, 0, 9, 9, 9, 9, 9, 9, }, 26 new int[] { 9, 9, 9, 0, 9, 9, 9, 0, 9, 0, 9, 9, 9, 9, 0, 9, 9, 0, 9, 9, 9, 9, 9, 0, 0, 0, 0, 0, 0, 0, 9, 0, 9, 0, 9, 9, 9, 9, 9, 9, }, 27 new int[] { 9, 0, 0, 0, 9, 9, 9, 0, 9, 0, 9, 0, 0, 0, 0, 9, 9, 0, 9, 9, 9, 9, 9, 0, 9, 9, 9, 9, 9, 9, 9, 0, 9, 0, 9, 0, 0, 0, 0, 9, }, 28 new int[] { 9, 0, 9, 9, 9, 9, 9, 0, 9, 0, 9, 9, 0, 9, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 9, 9, 9, 9, 9, 9, 9, 0, 9, 0, 9, 0, 9, 9, 9, 9, }, 29 new int[] { 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 9, 0, 9, 0, 9, 9, 0, 9, 9, 9, 9, 9, 0, 9, 0, 0, 0, 0, 0, 0, 0, 9, 0, 9, 0, 9, 9, 9, 9, }, 30 new int[] { 9, 0, 9, 9, 9, 9, 9, 9, 9, 0, 9, 9, 0, 9, 0, 9, 9, 0, 0, 9, 9, 9, 9, 0, 9, 0, 9, 9, 9, 9, 9, 9, 9, 0, 9, 0, 9, 0, 9, 9, }, 31 new int[] { 9, 0, 9, 9, 9, 9, 9, 9, 9, 0, 9, 0, 0, 9, 9, 9, 9, 9, 0, 9, 0, 0, 0, 0, 0, 0, 9, 9, 9, 9, 9, 9, 9, 0, 9, 0, 9, 0, 9, 9, }, 32 new int[] { 9, 0, 9, 9, 9, 9, 9, 9, 9, 0, 9, 9, 0, 0, 9, 9, 9, 9, 0, 9, 0, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 0, 9, 0, 9, 0, 9, 9, }, 33 new int[] { 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 9, 0, 9, 9, 9, 9, 9, 0, 9, 0, 9, 9, 9, 9, 9, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 9, 0, 9, 9, }, 34 new int[] { 9, 0, 9, 9, 9, 0, 9, 9, 9, 0, 9, 9, 0, 0, 0, 0, 0, 0, 0, 9, 0, 9, 9, 0, 9, 9, 0, 9, 9, 9, 9, 9, 0, 9, 9, 0, 0, 0, 9, 9, }, 35 new int[] { 9, 0, 9, 9, 9, 0, 9, 9, 9, 0, 9, 9, 9, 9, 0, 9, 0, 9, 0, 9, 0, 9, 9, 0, 9, 9, 0, 9, 9, 0, 0, 0, 0, 9, 9, 0, 9, 0, 9, 9, }, 36 new int[] { 9, 0, 9, 9, 9, 0, 9, 9, 9, 0, 0, 0, 9, 9, 0, 9, 0, 9, 9, 9, 0, 0, 9, 0, 0, 0, 0, 9, 9, 9, 0, 9, 9, 9, 9, 0, 9, 0, 9, 9, }, 37 new int[] { 9, 0, 9, 9, 9, 0, 9, 9, 9, 0, 9, 0, 9, 9, 0, 9, 0, 9, 9, 9, 0, 0, 9, 9, 0, 0, 9, 9, 0, 9, 0, 9, 9, 9, 9, 0, 9, 0, 9, 9, }, 38 new int[] { 9, 0, 0, 0, 9, 0, 0, 0, 9, 0, 9, 0, 9, 0, 0, 9, 0, 9, 9, 9, 9, 0, 9, 9, 9, 0, 9, 9, 0, 9, 9, 9, 0, 9, 9, 0, 9, 9, 9, 9, }, 39 new int[] { 9, 9, 9, 0, 9, 0, 9, 9, 9, 0, 9, 0, 9, 0, 9, 9, 0, 0, 0, 0, 9, 0, 9, 9, 9, 0, 9, 9, 0, 9, 9, 9, 0, 9, 9, 0, 9, 9, 9, 9, }, 40 new int[] { 9, 9, 9, 0, 9, 0, 9, 9, 9, 0, 9, 0, 9, 0, 9, 9, 9, 9, 9, 0, 9, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 9, 9, 9, }, 41 new int[] { 9, 0, 0, 0, 9, 0, 9, 0, 9, 0, 9, 9, 9, 0, 0, 9, 0, 0, 0, 0, 9, 0, 9, 9, 9, 0, 9, 9, 9, 9, 0, 9, 9, 9, 9, 0, 9, 9, 9, 9, }, 42 new int[] { 9, 0, 9, 9, 9, 0, 9, 0, 0, 0, 9, 9, 9, 9, 0, 9, 0, 9, 9, 9, 9, 0, 9, 9, 9, 0, 9, 9, 9, 9, 0, 9, 9, 9, 9, 0, 9, 9, 9, 9, }, 43 new int[] { 9, 0, 0, 0, 0, 0, 9, 9, 9, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 9, 9, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, }, 44 new int[] { 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, } 45 }; 46 Console.WriteLine("迷宫地图："); 47 PrintMap(); 48 System.Console.ReadKey(); 49 } 50 private static void PrintMap() { 51 Console.ForegroundColor = ConsoleColor.Green; 52 for (int i = 0; i < map.Length; i++) { 53 for (int j = 0; j < map.Length; j++) 54 if (map[i][j] == 9) 55 Console.Write("■"); 56 else 57 Console.Write(" ");//Console.Write("※"); 58 Console.WriteLine(); 59 } 60 } 61 }
先看看运行结果：
递归的方法跟之前的差不多，不过这次并不直接打印出来而只是做保存，把每一步的坐标点都记录下来
static List<List<string>> results = new List<List<string>>(); static List<string> way = new List<string>(); static void Go(int i, int j) { int m, n; map[i][j] = 1; if (i == endY && j == endX) { way.Clear(); for (m = 0; m < map.Length; m++) { for (n = 0; n < map.Length; n++) if (map[m][n] == 1) { way.Add("" + m + "," + n); } } results.Add((way.ToArray().Clone() as string[]).ToList<string>()); } if (map[i][j + 1] == 0) Go(i, j + 1); if (map[i + 1][j] == 0) Go(i + 1, j); if (map[i][j - 1] == 0) Go(i, j - 1); if (map[i - 1][j] == 0) Go(i - 1, j); map[i][j] = 0; }
然后在所有线路集合中找出最优线路也就是最短线路
List<string> bestWay = BestWayOut(); //找到最优解 MarkWayInMap(bestWay); //用最优解标记地图 Console.ReadKey(); StepByStep(startX, startY); //用字典记录坐标的步数[递归] WatchBestStep(); //输出最优步伐
static public List<string> BestWayOut() { Console.WriteLine(); Console.WriteLine(); Console.WriteLine(); Console.WriteLine("==================最优路线是=================="); var r = results.OrderBy(result => result.Count).First(); for (int i = 0; i < map.Length; i++) { for (int j = 0; j < map[i].Length; j++) { string str = i + "," + j; if (r.Contains(str)) { Console.ForegroundColor = ConsoleColor.Red; Console.Write("◆"); Console.ResetColor(); } else { switch (map[i][j]) { case 9: Console.ForegroundColor = ConsoleColor.Green; Console.Write("■"); Console.ResetColor(); break; default: Console.Write(" "); break; } } } Console.WriteLine(); } return r; } static public void MarkWayInMap(List<string> r) { for (int i = 0; i < map.Length; i++) { for (int j = 0; j < map[i].Length; j++) { string str = i + "," + j; if (r.Contains(str)) { map[i][j] = 1; } } } } static int count = 1;//用于记录坐标点的步数 static public void StepByStep(int x, int y) { map[x][y] = 2; dicStep[string.Format("{0},{1}", x, y)] = count++; if (x == endX && y == endY) { } else if (map[x][y + 1] == 1) { StepByStep(x, y + 1); } else if (map[x + 1][y] == 1) { StepByStep(x + 1, y); } else if (map[x][y - 1] == 1) { StepByStep(x, y - 1); } else if (map[x - 1][y] == 1) { StepByStep(x - 1, y); } } static public void WatchBestStep() { Console.WriteLine(); Console.WriteLine(); Console.WriteLine(); Console.WriteLine("==================最优路线走法=================="); for (int i = 0; i < map.Length; i++) { for (int j = 0; j < map[i].Length; j++) { string str = i + "," + j; if (dicStep.ContainsKey(str)) { Console.BackgroundColor = ConsoleColor.Red; Console.Write(string.Format("{0,2}", dicStep[str] % 10)); Console.ResetColor(); } else { switch (map[i][j]) { case 9: Console.ForegroundColor = ConsoleColor.Green; Console.Write("■"); Console.ResetColor(); break; default: Console.Write(" "); break; } } } Console.WriteLine(); } }
这个简单的40*40迷宫就有23280条线路，使用递归的速度还是比较慢的总共用时 接近7s，所以这个并不是最好的不过用来研究经典迷宫还是足够了，咱们只为了玩嘛是吧！
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