Eight

Time Limit: 10000/5000 MS (Java/Others) Memory Limit: 65536/32768 K (Java/Others)
Total Submission(s): 11243 Accepted Submission(s): 3022
Special Judge

Problem Description

The 15-puzzle has been around for over 100 years; even if you don't know it by that name, you've seen it. It is constructed with 15 sliding tiles, each with a number from 1 to 15 on it, and all packed into a 4 by 4 frame with one tile missing. Let's call the missing tile 'x'; the object of the puzzle is to arrange the tiles so that they are ordered as:


 1  2  3  4
 5  6  7  8
 9 10 11 12
13 14 15  x

where the only legal operation is to exchange 'x' with one of the tiles with which it shares an edge. As an example, the following sequence of moves solves a slightly scrambled puzzle:


 1  2  3  4     1  2  3  4     1  2  3  4     1  2  3  4
 5  6  7  8     5  6  7  8     5  6  7  8     5  6  7  8
 9  x 10 12     9 10  x 12     9 10 11 12     9 10 11 12
13 14 11 15    13 14 11 15    13 14  x 15    13 14 15  x
            r->            d->            r->

The letters in the previous row indicate which neighbor of the 'x' tile is swapped with the 'x' tile at each step; legal values are 'r','l','u' and 'd', for right, left, up, and down, respectively.

Not all puzzles can be solved; in 1870, a man named Sam Loyd was famous for distributing an unsolvable version of the puzzle, and
frustrating many people. In fact, all you have to do to make a regular puzzle into an unsolvable one is to swap two tiles (not counting the missing 'x' tile, of course).

In this problem, you will write a program for solving the less well-known 8-puzzle, composed of tiles on a three by three
arrangement.

Input

You will receive, several descriptions of configuration of the 8 puzzle. One description is just a list of the tiles in their initial positions, with the rows listed from top to bottom, and the tiles listed from left to right within a row, where the tiles are represented by numbers 1 to 8, plus 'x'. For example, this puzzle

1 2 3
x 4 6
7 5 8

is described by this list:

1 2 3 x 4 6 7 5 8

Output

You will print to standard output either the word ``unsolvable'', if the puzzle has no solution, or a string consisting entirely of the letters 'r', 'l', 'u' and 'd' that describes a series of moves that produce a solution. The string should include no spaces and start at the beginning of the line. Do not print a blank line between cases.

Sample Input

2 3 4 1 5 x 7 6 8

Sample Output

ullddrurdllurdruldr

　　题意：经典的八数码，给出当前状态，求出恢复到从小到大排列的顺序的任意一种路径，如果不存在就输出unsolvable。这一题用搜索解决。在看到其他人的题解看到这一题有很多种解法，这里我尝试的是用A*，第一次用A*→_→。

　　先说说解法，这一题需要先知道的知识：①对于八数码有解的情况，用当前的八个数字组成的序列，如果逆序对的对数是偶数的话就有解，否则无解，这是判断八数码有没有解的条件。②这里如果选择使用哈希记录状态的话，需要想个不错的方法。这里使用的方法是用康托展开。对于这东西可以自行百度一下。

　　这里使用A*，需要注意的是，我们选择扩展的状态是选择f(s)(估值函数)最小的状态。其中在计算g(s)(已使用代价)用的是每扩展一次就加一，h(s)(预估使用代价)是使用每一个位置到达正确位置的哈密顿距离之和来作为预估值。在将状态放入优先队列(OPEN表)的时候根据不同顺序比较g(s),h(s)效果可能不一样。

　　A*的本质就是每一次从还没有扩展的状态集合里面选出f(s)最小的那个状态来扩展，感觉看起来就是优先队列版的BFS。

　　当然，A*不一定就能提高效率，这就得看估值函数的选取了。

　　这一题码了两次，第一次是基本上跟着别人的代码写的，第二次自己写，但还是不得不稍微看一些自己写的代码，主要是对康托展开不是很熟悉。

上代码：

  1 #include <iostream>
  2 #include <cstdio>
  3 #include <cstring>
  4 #include <queue>
  5 #include <algorithm>
  6 #define MAX 400002
  7 using namespace std;
  8 typedef struct Node{
  9     int maze[3][3];
 10     int y,x;
 11     int hash;
 12     int h,g;
 13     bool isok(){
 14         return (0<=y && y<3 && 0<=x && x<3);
 15     }
 16 
 17     bool operator < (const Node a) const{
 18         if(h>a.h) return 1;
 19         if(h==a.h && g>a.g) return 1;
 20         return 0;
 21     }
 22 
 23 }Node;
 24 Node s;
 25 const int cantor[9] = {1,1,2,6,24,120,720,5040,40320};
 26 const int target = 322560;
 27 const int cy[]={0,0,1,-1};
 28 const int cx[]={1,-1,0,0};
 29 int vis[MAX];
 30 int pre[MAX];
 31 priority_queue<Node> p;
 32 bool check(Node c){
 33     int a[9];
 34     int k=0,sum=0;
 35     for(int i=0;i<3;i++) for(int j=0;j<3;j++) a[k++]=c.maze[i][j];
 36     for(int i=0;i<k;i++) for(int j=i+1;j<k;j++) if(a[i] && a[j] && a[i]>a[j]) sum++;
 37     return !(sum&1);
 38 }
 39 
 40 int getHash(Node c){
 41     int a[9];
 42     int k=0,h=0,sum=0;
 43     for(int i=0;i<3;i++) for(int j=0;j<3;j++) a[k++]=c.maze[i][j];
 44     for(int i=0;i<k;i++){
 45         sum=0;
 46         for(int j=0;j<i;j++){
 47             if(a[i]<a[j]) sum++;
 48         }
 49         h+=sum*cantor[i];
 50     }
 51     return h;
 52 }
 53 
 54 int getH(Node c){
 55     int res=0;
 56     for(int i=0;i<3;i++) for(int j=0;j<3;j++){
 57         res+=abs(i-(c.maze[i][j]-1)/3)+abs(j-(c.maze[i][j]-1)%3);
 58     }
 59     return res;
 60 }
 61 
 62 void astar(){
 63     while(!p.empty()) p.pop();
 64     memset(vis,-1,sizeof(vis));
 65     memset(pre,-1,sizeof(pre));
 66     Node u,v;
 67     s.h = getH(s);
 68     p.push(s);
 69     vis[s.hash]=-2;
 70     while(!p.empty()){
 71         u = p.top();
 72         p.pop();
 73         for(int i=0;i<4;i++){
 74             v=u;v.y+=cy[i];v.x+=cx[i];
 75             if(!v.isok()) continue;
 76             swap(v.maze[v.y][v.x],v.maze[u.y][u.x]);
 77             if(!check(v)) continue;
 78             v.hash = getHash(v);
 79             if(vis[v.hash]!=-1) continue;
 80             vis[v.hash]=i;  pre[v.hash]=u.hash;
 81             v.g++;
 82             v.h=getH(v);
 83             p.push(v);
 84             if(v.hash == target) return ;
 85 
 86         }
 87     }
 88 
 89 }
 90 
 91 bool scan(){
 92     char ch[100];
 93     if(!gets(ch)) return 0;
 94     int k=0,l=strlen(ch);
 95     for(int i=0;i<3;i++) for(int j=0;j<3;j++){
 96         while(k<l && ch[k]==' ') k++;
 97         if(ch[k]=='x'){
 98             s.maze[i][j]=0;
 99             s.y=i; s.x=j;
100         }
101         else s.maze[i][j]=ch[k]-'0';
102         k++;
103     }
104     return 1;
105 }
106 
107 void printPath(){
108     string ss;
109     ss.clear();
110     int next = target;
111     while(pre[next]!=-1){
112         if(vis[next]==0) ss+='r';
113         else if(vis[next]==1) ss+='l';
114         else if(vis[next]==2) ss+='d';
115         else ss+='u';
116         next = pre[next];
117     }
118     for(int i=(int)ss.length()-1;i>=0;i--) putchar(ss[i]);
119     puts("");
120 }
121 
122 int main()
123 {
124     //freopen("data.txt","r",stdin);
125     while(scan()){
126         if(!check(s)){
127             puts("unsolvable");
128         }else{
129             s.hash = getHash(s);
130             if(s.hash == target){
131                 puts("");
132             }else{
133                 astar();
134                 printPath();
135             }
136         }
137     }
138     return 0;
139 }

HDU 1043

  1 #include <iostream>
  2 #include <cstdio>
  3 #include <cstring>
  4 #include <algorithm>
  5 #include <queue>
  6 #define MAX 400002
  7 using namespace std;
  8 
  9 const int cantor[] = {1,1,2,6,24,120,720,5040,40320};
 10 const int target = 322560;
 11 int vis[MAX],pre[MAX];
 12 int cy[]={0,1,0,-1};
 13 int cx[]={1,0,-1,0};
 14 typedef struct Node{
 15     int maze[3][3];
 16     int h,g,y,x;
 17     int hash;
 18 
 19     bool isok(){
 20         return (0<=y && y<3 && 0<=x && x<3);
 21     }
 22 
 23     bool islegal(){
 24         int a[9],k=0,sum=0;
 25         for(int i=0;i<3;i++) for(int j=0;j<3;j++) a[k++]=maze[i][j];
 26         for(int i=0;i<k;i++) for(int j=i+1;j<k;j++) if(a[i] && a[j] && a[i]>a[j]) sum++;
 27         return !(sum&1);
 28     }
 29 
 30     bool operator < (const Node o)const{
 31         return h==o.h ? g>o.g : h>o.h;
 32     }
 33 }Node;
 34 Node s;
 35 priority_queue<Node> q;
 36 
 37 int getHash(Node e){
 38     int a[9],k=0,res=0;
 39     for(int i=0;i<3;i++) for(int j=0;j<3;j++) a[k++]=e.maze[i][j];
 40     for(int i=0;i<k;i++){
 41         int t=0;
 42         for(int j=0;j<i;j++) if(a[i]<a[j]) t++;
 43         res+=t*cantor[i];
 44     }
 45     return res;
 46 }
 47 
 48 int getH(Node e){
 49     int res=0;
 50     for(int i=0;i<3;i++) for(int j=0;j<3;j++){
 51         res+=abs(i-(e.maze[i][j]-1)/3)+abs(j-(e.maze[i][j]-1)%3);
 52     }
 53     return res;
 54 }
 55 
 56 void astar(){
 57     Node u,v;
 58     while(!q.empty()) q.pop();
 59     memset(vis,-1,sizeof(vis));
 60     memset(pre,-1,sizeof(pre));
 61     s.g=0;
 62     s.h=getH(s);
 63     q.push(s);
 64     vis[s.hash]=-2;
 65     while(!q.empty()){
 66         u=q.top();
 67         q.pop();
 68         //cout<<u.hash<<endl;
 69         for(int i=0;i<4;i++){
 70             v=u;v.y+=cy[i];v.x+=cx[i];
 71             if(v.isok()){
 72                 swap(v.maze[v.y][v.x],v.maze[u.y][u.x]);
 73                 if(v.islegal()){
 74                     v.hash = getHash(v);
 75                     if(vis[v.hash]==-1){
 76                         vis[v.hash]=i;
 77                         pre[v.hash]=u.hash;
 78                         v.g++;  v.h=getH(v);
 79                         q.push(v);
 80                         if(v.hash == target) return ;
 81                     }
 82                 }
 83 
 84             }
 85         }
 86     }
 87 }
 88 
 89 void printPath(){
 90     string ss="";
 91     int next=target;
 92     while(pre[next]!=-1){
 93         switch(vis[next]){
 94             case 0:ss+='r';break;
 95             case 1:ss+='d';break;
 96             case 2:ss+='l';break;
 97             case 3:ss+='u';break;
 98         }
 99         next=pre[next];
100     }
101     for(int i=(int)ss.length()-1;i>=0;i--) putchar(ss[i]);
102     putchar('
');
103 }
104 
105 bool get(){
106     char ss[100];
107     if(gets(ss)){
108         int k=0,l=strlen(ss);
109         for(int i=0;i<3;i++) for(int j=0;j<3;j++){
110             while(k<l && ss[k]==' ') k++;
111             if(ss[k]=='x'){
112                 s.maze[i][j]=0;
113                 s.y=i; s.x=j;
114             }else s.maze[i][j]=ss[k]-'0';
115             k++;
116         }
117         return 1;
118     }
119     return 0;
120 
121 }
122 
123 int main()
124 {
125     //freopen("data.txt","r",stdin);
126     while(get()){
127         if(s.islegal()){
128             s.hash = getHash(s);
129             if(s.hash == target) puts("");
130             else{
131                 astar();
132                 printPath();
133             }
134         }else puts("unsolvable");
135     }
136     return 0;
137 }

POJ 1077