简介
- 实验项目 2: 栈结构及其应用
- 实验题目: 迷宫问题求解
- 实验内容:
一个迷宫可以看成是由 m × n 个房间组成的矩形,迷宫内部的每个房间有 4个方向,每个方向或者有障碍(如墙)而不能通过,或者无障碍而能通过。 入口为左上角房间,出口为右下角房间,问是否有简单路径从入口到出口,若有则输出一条这样的路径;否则,提示迷宫无入口到出口路经。 - 实验要求:
- 设计一个迷宫及其障碍的表示方式,并能随机或手动生成迷宫,并以适当方式展示。
- 设计并实现一个非递归的算法,输出从入口到出口的一条路径(如存在)。
- 设计并实现一个递归的算法,找出从入口到出口的一条路径(如存在)。
- 选做:如果有多条路径,设计并实现一个算法找到步数最少的路径(捷径)。
- 选做:如果有多条路径,设计并实现一个算法找到所有路径。
- 以适当的方式展示迷宫和所走路径
作业内容三选一,感觉迷宫还行,可以做做,发现迷宫的恰当表示和随机生成有学问啊
迷宫生成
迷宫生成是将迷宫全部初始化为墙,然后打通墙,制造迷宫的过程
基本知识
迷宫生成主要有四种方法
递归回溯算法
- 深度优先搜索,递归地打通未打通的区域
- 思想简单,但生成的迷宫通路十分明显
递归分割算法
- 又名十字递归算法,递归地将地图分为四个房间,然后联通四个房间
- 生成的迷宫就是像一个一个房间,适合RPG游戏
随机Prim算法
- 最小生成树算法,从已有的通路开始每次随机地选择一个方向打通迷宫
- 通路不明显,适合迷宫游戏
Kruskal算法
- 最小生成树算法,利用并查集随机地选择墙打通
这篇博客有动图可以帮助理解 迷宫生成
Prim算法
这里使用Prim算法
- 初始化迷宫全为墙。
- 选一个是迷宫通路的格子,然后随机选择它的邻墙。一般一开始的时候是起点(左上角)。
- 随机选择墙时:
- 如果它联通的格子不是迷宫的通路:
- 把墙打通。
- 自然那个格子变成了迷宫的通路.。
- 如果它联通的格子已经是通路了,选择其他格子去考虑邻墙。
这里的随机使一般的最小生成树Prim算法有所区别,该如何随机选择墙打通呢?
随机化种子
void srand(unsigned int seed) 播种由函数 rand 使用的随机数发生器。
int rand(void) 返回一个范围在 0 到 RAND_MAX 之间的伪随机数。
使用时间作为种子
srand(time(NULL));
srand ((int) time ((time_t *) NULL));
迷宫表示
设迷宫宽为width,高为height
表示迷宫一般是使用 \(width * height\) 大小的矩阵来表示,一个单元表示迷宫的通路与否。
如 11 * 4
.*****...**..**...*.*.*.*..*.*.*.*....***...
.表示通路
*或者#表示障碍
但是为了更好的表示迷宫,表示迷宫与通路的关系,方便我们观看
使用 墙 + 单元 的方式来表示迷宫
则还需一圈外围围住迷宫的围墙
四方向
可以上下左右的走
总共需要 \((2*width + 1)*(2*height + 1)\) 大小的内存来表示
如 4 * 2
+-+-+-+-+| | | | |+-+-+-+-+| | | | |+-+-+-+-+
空格为一个迷宫单元,其他为墙
八方向
可以上下左右,且对角线的走
这样可以表示单元的上下左右与对角线的通路情况,虽然丑
很明显,一个九宫格,周围八个都是墙,中间是迷宫的单元
总共需要\((3 * height) * (3 * width)\) 大小的内存
如3 * 3
/=\/=\/=\| || || |\=/\=/\=//=\/=\/=\| || || |\=/\=/\=//=\/=\/=\| || || |\=/\=/\=/
迷宫解法
主要有三种解法
- 深度优先搜索 DFS
- 广度优先搜索 BFS
- 启发式算法 A*
这里使用的深度优先搜索 ( DFS )
一条路走到黑,走不动就返回,直到走到终点
DFS就不作过多陈述
代码及运行结果迷宫数据类型定义
- 采用结构体,中有6/10个变量
- 4/8个方向,上下左右,(左上左下右上右下),表示该单元墙的打通情况
- Visited变量,在DFS记录是否走过
- Path变量,在DFS中记录是否为解法通路
四方向
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+.+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+|. | | | | |....... | |...| | | |+.+-+-+-+ + +-+ + +-+-+ + +-+ + +-+.+-+-+.+ + +-+ +.+.+ + +-+ +-+-+-+-+-+-+ + +|...|...| | | | | | |...|.....| | | |.|...| | | | | |+-+.+.+.+ +-+ + +-+-+ +-+-+-+-+-+.+ +.+-+-+ +-+ +-+.+-+.+-+-+-+ + + +-+-+-+ + +|...|.|.| | | | | | |...| |.....| | |...|...| | | |.....| |+.+-+.+.+-+ + + + + +-+ + + + +.+-+-+-+-+.+-+ + +.+-+.+-+ +-+ +-+ + +.+-+.+-+-+|.....|.| | | | | | | |. |.....| | |.| |... | | | |.|...|...|+ +-+-+.+-+-+-+ +-+ + + +-+-+-+.+-+ +.+-+-+ +-+-+.+ + +.+-+-+ + +-+ +.+.+-+.+.+| |.......| | | |.........| | |. |.....| | |.....| | |.|...|.|.|+-+-+-+-+-+-+.+ + +-+-+.+-+-+-+-+ + +.+-+-+-+.+-+-+ + +-+-+.+-+-+-+-+.+-+.+.+.+| .| |.....|...| | |.......|.| | | |.........|.| |...|.|+ +-+-+-+-+-+.+-+-+.+-+-+.+.+-+-+ + + +-+-+.+.+ + + +-+-+ +-+-+-+-+.+.+ + +-+.+| | |.....|.......|.| | | |...| | | | | |...|.| | |.|+-+-+ + +-+-+-+-+.+-+-+ +-+.+ + +-+-+-+-+-+-+-+ +-+-+ + +-+-+ + +.+-+.+ +-+ +.+| | | | |.....| |...| | | | | | | | |.....| |.....|+-+ + + + + + + +-+-+.+-+.+-+ +-+-+ +-+-+ + +-+-+ + +-+-+-+ + + +-+-+-+ +.+-+-+| | | | | | | |...|...| | | | | | | | | |.| |+ +-+ + +-+-+-+ + + +-+.+-+.+-+-+ + +-+ + +-+-+-+ + +-+ +-+-+-+-+ +-+-+-+.+-+ +| | | | | |...|...| | | | | | | | | | |...| |+-+-+-+ +-+ +-+-+ + +-+ +.+-+.+ +-+-+ + +-+-+ + +-+-+-+-+ + + + + + + +-+-+.+ +| | | |..... | | | | | | |...|+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+.+
点击查看代码
#include #include #include #define WIDTH 39 #define HEIGHT 11#define UP 0#define RIGHT 1#define DOWN 2#define LEFT 3#ifdef TRUE#undef TRUE#endif /* TRUE */#define TRUE 1#define cell_empty(a) (!(a)->up && !(a)->right && !(a)->down && !(a)->left)typedef struct { unsigned int up : 1;//占一位 unsigned int right : 1; unsigned int down : 1; unsigned int left : 1; unsigned int path : 1; unsigned int visited : 1;}cell;typedef cell * maze_p;void CreateMaze (maze_p maze, int width, int height);void SolveMaze (maze_p maze, int width, int height);void PrintMaze (maze_p maze, int width, int height);int SolveMazeRec (maze_p maze, maze_p mp, int width, int height);int main (int argc, char *argv []){ int width = WIDTH; int height = HEIGHT; maze_p maze; if (argc >= 2) width = atoi (argv [1]);//atoi函数, 将一个字符串转化为整数 if (argc >= 3) height = atoi (argv [2]); if (argc >= 4) srand (atoi (argv [3]));//随机种子 else srand ((int) time ((time_t *) NULL));// 用系统时初始化随机种子 if (width <= 0 || height = maze && cell_empty (mp - width)) paths [directions ++] = UP; if (mp < maze_pop && ((mp - maze + 1) % width) && cell_empty (mp + 1))//判断是不是最右 paths [directions ++] = RIGHT; if ((mp + width) maze && ((mp - maze) % width) && cell_empty (mp - 1)) //判断是不是最左 paths [directions ++] = LEFT;//在mp可选择的路中随机一个 if (directions) { visits--; directions = ((unsigned) rand () % directions); switch (paths [directions]) { case UP: mp->up = TRUE;//标记这个cell向上走 (mp -= width)->down = TRUE;//相反,走过去的cell标记为向下走 break; case RIGHT: mp->right = TRUE; (++mp)->left = TRUE; break; case DOWN: mp->down = TRUE; (mp += width)->up = TRUE; break; case LEFT: mp->left = TRUE; (--mp)->right = TRUE; break; default: break; } } else //没有可走的cell { do { if (++mp > maze_pop)//超过了就回到起点 mp = maze; } while (cell_empty (mp)); // 找到一个已被打通的cell } }}/* CreateMaze */int SolveMazeRec (maze_p maze, maze_p mp, int width, int height){mp->visited = TRUE;if(mp == (maze + (width * height) - 1)){mp->path = TRUE;return 0;}for(int sel = UP; sel up && !(mp - width)->visited){if( ! SolveMazeRec (maze, mp - width, width, height) ){mp->path = TRUE;return 0;}}break;case RIGHT:if (mp->right && !(mp + 1)->visited){if( ! SolveMazeRec (maze, mp + 1, width, height) ){mp->path = TRUE;return 0;}}break;case DOWN:if (mp->down && !(mp + width)->visited){if( ! SolveMazeRec (maze, mp + width, width, height) ){mp->path = TRUE;return 0;}}break;case LEFT:if (mp->left && !(mp - 1)->visited){if( ! SolveMazeRec (maze, mp - 1, width, height) ){mp->path = TRUE;return 0;}}break;default:break;}}return 1;}void SolveMaze (maze_p maze, int width, int height){ maze_p *stack, mp = maze; int sp = 0; stack = (maze_p *) calloc (width * height, sizeof (maze_p)); if (stack == NULL) { (void) fprintf (stderr, "Cannot allocate memory!\n"); exit (EXIT_FAILURE); }// 起点已访问 (stack [sp++] = mp)->visited = TRUE; while (mp != (maze + (width * height) - 1)) //没到终点 { if (mp->up && !(mp - width)->visited)//可走上,上没去过 stack [sp++] = mp - width; if (mp->right && !(mp + 1)->visited) stack [sp++] = mp + 1; if (mp->down && !(mp + width)->visited) stack [sp++] = mp + width; if (mp->left && !(mp - 1)->visited) stack [sp++] = mp - 1; if (stack [sp - 1] == mp) --sp;//如果走到头了,那就回去一步 (mp = stack [sp - 1])->visited = TRUE;//两步 } while (sp--)//遍历一遍,标记为路径 if (stack [sp]->visited) stack [sp]->path = TRUE; free (stack);}/* SolveMaze */void PrintMaze (maze_p maze, int width, int height){ int w, h; char *line, *lp; line = (char *) calloc ((width + 1) * 2, sizeof (char)); if (line == NULL) { (void) fprintf (stderr, "Cannot allocate memory!\n"); exit (EXIT_FAILURE); } maze->up = TRUE; (maze + (width * height) - 1)->down = TRUE; // 第一行 for (lp = line, w = 0; w up)//如果为出迷宫路径,则为 . ,否则为 空 *lp++ = ((maze + w)->path) ? '.' : ' '; else *lp++ = '-'; } //一行 长 2*width + 1 *lp++ = '+'; (void) puts (line); //system("pause"); for (h = 0; h < height; h++)// { for (lp = line, w = 0; w left) *lp++ = ((maze + w)->path && (maze + w - 1)->path) ? '.' : ' '; else *lp++ = '|';//墙 *lp++ = ((maze + w)->path) ? '.' : ' '; } *lp++ = '|'; (void) puts (line); for (lp = line, w = 0; w down) *lp++ = ((maze + w)->path && (h == height - 1 || (maze + w + width)->path)) ? '.' : ' '; else *lp++ = '-'; } *lp++ = '+'; (void) puts (line); maze += width; } free (line);}/* PrintMaze */
八方向
/ \/=\/=\/=\/=\/=\/=\/=\/=\/=\/=\| || || || || || || || |\=/ =/\ /\= \=/\=/ =/\ / /\=/\ //= /=\/ \/=\ =\/= /=\/ / \/=\/ \| || || || || || || |\ \ /\=/\ \=/\=/\=/\=/\=/\ /\ // \ \/=\/ \ =\/=\/=\/=\/=\/ \/ \| || || || || || || || || |\= \=/ / =/\=/\ \=/ =/\= = \=//=\ = / /=\/=\/ \ = /=\/= =\ =\| || || || || || || || |\= = \= =/\ /\ \= \=/\=/\= \ //= =\ = =\/ \/ \ =\ =\/=\/=\ \| || || || || || || || || |\=/\=/\=/\=/\=/\=/\=/\=/\=/\=/\ /
/.\/=\/=\/=\/=\/=\/=\/=\/=\/=\/=\|....||....|| || ||....||.|| || |\=/.=/\./\=.\=/\=/.=/\./../\=/\ //=./=\/.\/=\.=\/=./=\/../.\/=\/ \|. ||....||....|| ||.||....|| |\. \ /\=/\. \=/\=/\=/\=/\=/\./\ //.\ \/=\/.\ =\/=\/=\/=\/=\/.\/ \|.|| ||.||.|| || || ||....||. |\=.\=/../.=/\=/\ \=/.=/\=..= \=//=\.=./../=\/=\/ \ =./=\/=..=\ =\| ||.||.|| || ||.......||.|| |\= = \= =/\ /\ \= \=/\=/\=.\ //= =\ = =\/ \/ \ =\ =\/=\/=\. \| || || || || || || || ||.|\=/\=/\=/\=/\=/\=/\=/\=/\=/\=/\./
点击查看代码
/* * @Author: Az1r * @Date: 2022-09-30 20:47:13 */#include #include #include #define WIDTH 9#define HEIGHT 5#define UP 0#define RIGHT 1#define DOWN 2#define LEFT 3#define UL 4#define UR 5#define DL 6#define DR 7#ifdef TRUE#undef TRUE#endif /* TRUE */#define TRUE 1#define cell_empty(a) (!(a)->up && !(a)->right && !(a)->down && !(a)->left && !(a)->ul && !(a)->ur && !(a)->dl && !(a)->dr)typedef struct { unsigned int up : 1;//表示占1位 unsigned int right : 1; unsigned int down : 1; unsigned int left : 1; unsigned int ul : 1; unsigned int ur : 1; unsigned int dl : 1; unsigned int dr : 1; unsigned int path : 1; unsigned int visited : 1;}cell;typedef cell * maze_p;void CreateMaze (maze_p maze, int width, int height);void PrintMaze (maze_p maze, int width, int height); int SolveMazeRec (maze_p maze, maze_p mp, int width, int height);//递归版void SolveMaze (maze_p maze, int width, int height);//非递归版int main (int argc, char *argv []){ int width = WIDTH; int height = HEIGHT; int select = 0; maze_p maze; if (argc >= 2) width = atoi (argv [1]);//atoi函数,字符串转整数 if (argc >= 3) height = atoi (argv [2]); if (argc >= 4) srand (atoi (argv [3]));//随机种子 else srand ((int) time ((time_t *) NULL));//以时间作为随机种子if(argc >= 5){select = atoi (argv [4]);} if (width <= 0 || height = maze && cell_empty (mp - width))//若可以打通上面的墙,下同理 paths [directions ++] = UP; if (mp < maze_pop && ((mp - maze + 1) % width) && cell_empty (mp + 1)) paths [directions ++] = RIGHT; if ((mp + width) maze && ((mp - maze) % width) && cell_empty (mp - 1)) paths [directions ++] = LEFT; if ((mp - 1 - width) >= maze && ((mp - maze) % width) && mp > (maze + width - 1) && cell_empty (mp - 1 - width)) paths [directions ++] = UL; if ((mp + 1 - width) > maze && ((mp - maze + 1) % width) && mp > (maze + width - 1) && cell_empty (mp + 1 - width)) paths [directions ++] = UR; if ((mp - 1 + width) < maze_pop && ((mp - maze) % width) && mp <= (maze_pop - width) && cell_empty (mp - 1 + width)) paths [directions ++] = DL; if ((mp + 1 + width) <= maze_pop && ((mp - maze + 1) % width) && mp up = TRUE; //该单元上墙打通 (mp -= width)->down = TRUE;//该单元上面的单元,下墙打通 break; //以下同理 case RIGHT: mp->right = TRUE; (++mp)->left = TRUE; break; case DOWN: mp->down = TRUE; (mp += width)->up = TRUE; break; case LEFT: mp->left = TRUE; (--mp)->right = TRUE; break; case UL: mp->ul = TRUE; (mp -= width + 1)->dr = TRUE; break; case UR: mp->ur = TRUE; (mp -= width - 1)->dl = TRUE; break; case DL: mp->dl = TRUE; (mp += width - 1)->ur = TRUE; break; case DR: mp->dr = TRUE; (mp += width + 1)->ul = TRUE; break; default: break; } } else //没有符合条件的墙{ do { if (++mp > maze_pop)//超过了出口,就再从入口找起 mp = maze; } while (cell_empty (mp)); // 符合条件 } }}/* CreateMaze */int SolveMazeRec (maze_p maze, maze_p mp, int width, int height){mp->visited = TRUE;if(mp == (maze + (width * height) - 1))//可以走到出口就是0{mp->path = TRUE;return 0;}for(int sel = UP; sel up && !(mp - width)->visited)//可以走上,下同{if( ! SolveMazeRec (maze, mp - width, width, height) ){mp->path = TRUE;return 0;}}break;case RIGHT:if (mp->right && !(mp + 1)->visited){if( ! SolveMazeRec (maze, mp + 1, width, height) ){mp->path = TRUE;return 0;}}break;case DOWN:if (mp->down && !(mp + width)->visited){if( ! SolveMazeRec (maze, mp + width, width, height) ){mp->path = TRUE;return 0;}}break;case LEFT:if (mp->left && !(mp - 1)->visited){if( ! SolveMazeRec (maze, mp - 1, width, height) ){mp->path = TRUE;return 0;}}break;case UL:if (mp->ul && !(mp - 1 - width)->visited){if( ! SolveMazeRec (maze, mp - 1 - width, width, height)){mp->path = TRUE;return 0;}}break;case UR:if (mp->ur && !(mp + 1 - width)->visited){if( ! SolveMazeRec (maze, mp + 1 - width, width, height)){mp->path = TRUE;return 0;}}break;case DL:if (mp->dl && !(mp - 1 + width)->visited){if (! SolveMazeRec (maze, mp - 1 + width, width, height)){mp->path = TRUE;return 0;}}break;case DR:if (mp->dr && !(mp + 1 + width)->visited){if (! SolveMazeRec (maze, mp + 1 + width, width, height)){mp->path = TRUE;return 0;}}break;default:break;}}return 1;}void SolveMaze (maze_p maze, int width, int height) { maze_p *stack, mp = maze; int sp = 0; stack = (maze_p *) calloc (width * height, sizeof (maze_p)); if (stack == NULL) { (void) fprintf (stderr, "Cannot allocate memory!\n"); exit (EXIT_FAILURE); }//入口走过了 (stack [sp++] = mp)->visited = TRUE; while (mp != (maze + (width * height) - 1)) //出口{ if (mp->up && !(mp - width)->visited)//判断是否可走,且是否走过;下面同理stack [sp++] = mp - width; if (mp->right && !(mp + 1)->visited) stack [sp++] = mp + 1; if (mp->down && !(mp + width)->visited) stack [sp++] = mp + width; if (mp->left && !(mp - 1)->visited) stack [sp++] = mp - 1; if (mp->ul && !(mp - 1 - width)->visited) stack [sp++] = mp - 1 - width; if (mp->ur && !(mp + 1 - width)->visited) stack [sp++] = mp + 1 -width; if (mp->dl && !(mp - 1 + width)->visited) stack [sp++] = mp - 1 + width; if (mp->dr && !(mp + 1 + width)->visited) stack [sp++] = mp + 1 + width; if (stack [sp - 1] == mp) --sp;//无路可走,那就回退 (mp = stack [sp - 1])->visited = TRUE;//这其实是两步 } while (sp--)//回退栈,栈中保存的符合条件的即为路径 if (stack [sp]->visited) stack [sp]->path = TRUE; free (stack);}/* SolveMaze */void PrintMaze (maze_p maze, int width, int height){ int w, h; char *line, *lp; line = (char *) calloc ((width + 1) * 3, sizeof (char)); if (line == NULL) { (void) fprintf (stderr, "Cannot allocate memory!\n"); exit (EXIT_FAILURE); } maze->up = TRUE; (maze + (width * height) - 1)->down = TRUE; //system("pause"); for (h = 0; h < height; h++){for (lp = line, w = 0; w ul)//若墙通,则要么走过,要么没走 *lp++ = ((maze + w)->path && (maze + w - 1 - width)->path) ? '.' : ' '; else //墙不通则打印墙 *lp++ = '/'; if ((maze + w)->up) *lp++ = ((maze + w)->path && ((maze + w - width)->path || h == 0) )? '.' : ' '; else *lp++ = '='; if ((maze + w)->ur) *lp++ = ((maze + w)->path && (maze + w + 1 - width)->path) ? '.' : ' '; else { *lp++ = '\\';} } (void) puts (line); for (lp = line, w = 0; w left) *lp++ = ((maze + w)->path && (maze + w - 1)->path) ? '.' : ' '; else *lp++ = '|'; *lp++ = ((maze + w)->path) ? '.' : ' '; if ((maze + w)->right) *lp++ = ((maze + w)->path && (maze + w + 1)->path) ? '.' : ' '; else *lp++ = '|'; } (void) puts (line); for (lp = line, w = 0; w dl) *lp++ = ((maze + w)->path && (maze + w - 1 + width)->path) ? '.' : ' '; else { *lp++ = '\\';} if ((maze + w)->down) *lp++ = ((maze + w)->path && ( (maze + w + width)->path || (h + 1) == height) ) ? '.' : ' '; else *lp++ = '='; if ((maze + w)->dr) *lp++ = ((maze + w)->path && (maze + w + 1 + width)->path) ? '.' : ' '; else { *lp++ = '/';} } (void) puts (line); maze += width; } free (line);}/* PrintMaze */
参考博客
Random maze-generator FAQ
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