poj 1077 & hdu 1043 Eight （ 多种解法：预处理、bfs、dbfs、IDA*、A*）
Eight
Time Limit: 10000/5000 MS (Java/Others) Memory Limit: 65536/32768 K (Java/Others)
Total Submission(s): 9114 Accepted Submission(s): 2450
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:
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:
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.
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
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
ps：八数码有很多种解法 先贴一种用空间换时间的方法
pps：做八数码的前提：会基础的bfs 了解康拓判重 知道八数码的可解性
思路一：
因为内存还是给的蛮多的 八数码的情况又不是蛮多（不像蛋疼的立体八数码） 而且最终状态只有一种 所以完全可以把答案全部存下来 这种方法还不用判断八数码是否可解 还是蛮好的一种方法 不过只适合多组输入 不适合单组输入 hdu上能过 poj上就不能过了 所以还是得写其他的代码呀 
代码：
//Memory 20908K Time 687MS
#include <iostream>
#include <cstdio>
#include <cstring>
#include <string>
#include <cmath>
#include <queue>
#define maxn 370000
using namespace std;
int fac[15]= {40320,5040,720,120,24,6,2,1,1}; //康拖展开判重
int vis[maxn];
int cnt=0;
string anspath[maxn],ans;
int dx[4]={-1,1,0,0}; // up down left right
int dy[4]={0,0,-1,1};
char dir[5]="durl";
struct Tnode
{
char s[9];
int loc; //“0”的位置
int val; //康拖展开的值
string path; //路径
} cur,now;
queue<Tnode> q;
int kangtuo(char *ss) // 求康拓展开的值
{
int i,j,t,sum=0;
for(i=0; i<9; i++)
{
t=0;
for(j=i+1; j<9; j++)
{
if(ss[i]>ss[j]) t++;
}
sum+=t*fac[i];
}
return sum+1;
}
void bfs()
{
int i,j,t,t1;
int tempv,x,y,nx,ny;
char temps[9];
memset(vis,0,sizeof(vis));
while(!q.empty()) q.pop();
for(i=0;i<8;i++)
cur.s[i]=i+1-0+'0';
cur.s[8]='0';
cur.loc=9;
cur.path="";
cur.val=kangtuo(cur.s);
vis[cur.val]=1;
anspath[cur.val]=cur.path;
q.push(cur);
while(!q.empty())
{
now=q.front();
t=now.loc;
y=t%3;
if(y==0) y=3;
x=(t-y)/3+1;
q.pop();
for(i=0;i<4;i++)
{
nx=x+dx[i];
ny=y+dy[i];
t1=(nx-1)*3+ny;
memcpy(temps,now.s,sizeof(temps));
temps[t-1]=now.s[t1-1];
temps[t1-1]='0';
tempv=kangtuo(temps);
if(nx>=1&&nx<=3&&ny>=1&&ny<=3&&!vis[tempv])
{
vis[tempv]=1;
cur.loc=t1;
cur.path=now.path+dir[i];
memcpy(cur.s,temps,sizeof(temps));
cur.val=tempv;
q.push(cur);
anspath[cur.val]=cur.path; // 每次将情况存下来
}
}
}
}
int main()
{
int i,j,temp1;
char c;
bfs(); // 从结果倒着搜 将各种情况存下来
while(cin>>c)
{
if(c=='x')
{
cur.loc=1;
cur.s[0]='0';
}
else cur.s[0]=c;
for(i=1; i<9; i++)
{
cin>>c;
if(c=='x')
{
cur.loc=i+1;
cur.s[i]='0';
}
else cur.s[i]=c;
}
cur.val=kangtuo(cur.s);
if(vis[cur.val]) // 若存在则倒着输出路径就够了
{
temp1=anspath[cur.val].length();
for(i=temp1-1;i>=0;i--)
printf("%c",anspath[cur.val][i]);
}
else printf("unsolvable");
printf("\n");
}
return 0;
}
思路二：
单项bfs 将路径存下来 递归输出答案 记住最好不用string memcpy STL 这样会快一点
代码：
// Memory 5548K Time 344MS
#include <iostream>
#include <cstdio>
#include <cstring>
#include <string>
#include <cmath>
#include <queue>
#define maxn 370000
using namespace std;
int ans;
int fac[]= {40320,5040,720,120,24,6,2,1,1}; //康拖展开判重
bool vis[maxn];
int dx[]= {-1,1,0,0}; // up down left right
int dy[]= {0,0,-1,1};
char dir[]="udlr";
struct Tnode
{
char s[9];
int loc; // “0”的位置
int val; // 康拖展开的值
char d; // 方向
int fa;
} cur,now,q[maxn];
bool judge(char *ss) // 根据逆序数是否为奇数判断是否有解
{
int i,j,t=0;
for(i=0; i<9; i++)
{
if(ss[i]=='0') continue ;
for(j=i+1; j<9; j++)
{
if(ss[i]>ss[j]&&ss[j]!='0') t++;
}
}
if(t&1) return false ;
return true ;
}
int kangtuo(char *ss) // 求康拓展开的值
{
int i,j,t,sum=0;
for(i=0; i<9; i++)
{
t=0;
for(j=i+1; j<9; j++)
{
if(ss[i]>ss[j]) t++;
}
sum+=t*fac[i];
}
return sum+1;
}
bool bfs()
{
int i,j,t,t1;
int head=0,tail=-1;
int tempv,x,y,nx,ny,nval,npos;
char temps[9];
memset(vis,0,sizeof(vis));
cur.fa=-1;
cur.val=kangtuo(cur.s);
vis[cur.val]=1;
q[++tail]=cur;
while(head<=tail)
{
now=q[head];
t=now.loc;
nval=now.val;
if(nval==46234)
{
ans=head;
return true ;
}
y=t%3;
if(y==0) y=3;
x=(t-y)/3+1;
for(i=0; i<4; i++)
{
nx=x+dx[i];
ny=y+dy[i];
if(!(nx>=1&&nx<=3&&ny>=1&&ny<=3)) continue ;
t1=(nx-1)*3+ny;
for(j=0; j<9; j++)
{
temps[j]=now.s[j];
}
temps[t-1]=now.s[t1-1];
temps[t1-1]='0';
tempv=kangtuo(temps);
if(!vis[tempv])
{
vis[tempv]=1;
cur.loc=t1;
cur.d=dir[i];
cur.fa=head;
for(j=0; j<9; j++)
{
cur.s[j]=temps[j];
}
cur.val=tempv;
q[++tail]=cur;
}
}
head++;
}
return false ;
}
void output(int k) // 递归输出答案
{
if(k==0) return ;
else
{
output(q[k].fa);
printf("%c",q[k].d);
}
}
int main()
{
int i,j,temp1,vv;
char c;
while(cin>>c)
{
if(c=='x')
{
cur.loc=1;
cur.s[0]='0';
}
else cur.s[0]=c;
for(i=1; i<9; i++)
{
cin>>c;
if(c=='x')
{
cur.loc=i+1;
cur.s[i]='0';
}
else cur.s[i]=c;
}
if(judge(cur.s)&&bfs())
{
output(ans);
printf("\n");
}
else printf("unsolvable\n");
}
return 0;
}
思路三：
双向bfs 基本思路和单项一样 就是注意从后往前搜存方向得反过来存 两个递归输出答案也有一点点区别 因为我用的双向bfs采用递归输出答案 所以得建两个映射记录每个节点的父节点对应的下标
代码：
// Memory 4040K Time 32MS
#include <iostream>
#include <cstdio>
#include <cstring>
#include <string>
#include <cmath>
#include <queue>
#define maxn 370000
using namespace std;
int ans1,ans2;
int fac[]= {40320,5040,720,120,24,6,2,1,1}; //康拖展开判重
bool vis1[maxn],vis2[maxn];
int dx[]= {-1,1,0,0}; // up down left right
int dy[]= {0,0,-1,1};
int mp1[maxn],mp2[maxn];
char dir1[]="udlr";
char dir2[]="durl";
struct Tnode
{
char s[9];
int loc; // “0”的位置
int val; // 康拖展开的值
char d; // 方向
int fa;
int step;
} cur,now,q1[maxn],q2[maxn];
bool judge(char *ss) // 根据逆序数是否为奇数判断是否有解
{
int i,j,t=0;
for(i=0; i<9; i++)
{
if(ss[i]=='0') continue ;
for(j=i+1; j<9; j++)
{
if(ss[i]>ss[j]&&ss[j]!='0') t++;
}
}
if(t&1) return false ;
return true ;
}
int kangtuo(char *ss) // 求康拓展开的值
{
int i,j,t,sum=0;
for(i=0; i<9; i++)
{
t=0;
for(j=i+1; j<9; j++)
{
if(ss[i]>ss[j]) t++;
}
sum+=t*fac[i];
}
return sum;
}
bool bfs()
{
int i,j,t,t1,step1,step2;
int head1=0,tail1=-1,head2=0,tail2=-1;
int tempv,x,y,nx,ny,nval,npos;
char temps[9];
memset(vis1,0,sizeof(vis1));
memset(vis2,0,sizeof(vis2));
cur.step=0;
cur.val=kangtuo(cur.s);
vis1[cur.val]=1;
mp1[cur.val]=0;
q1[++tail1]=cur;
strcpy(cur.s,"123456780");
cur.val=kangtuo(cur.s);
cur.loc=9;
vis2[cur.val]=1;
mp2[cur.val]=0;
q2[++tail2]=cur;
step1=step2=0;
while(1)
{
while(q1[head1].step<=step1&&head1<=tail1)
{
now=q1[head1];
t=now.loc;
nval=now.val;
if(vis2[nval])
{
ans1=head1;
ans2=mp2[nval];
return true ;
}
y=t%3;
if(y==0) y=3;
x=(t-y)/3+1;
for(i=0; i<4; i++)
{
nx=x+dx[i];
ny=y+dy[i];
if(!(nx>=1&&nx<=3&&ny>=1&&ny<=3)) continue ;
t1=(nx-1)*3+ny;
for(j=0; j<9; j++)
{
temps[j]=now.s[j];
}
temps[t-1]=now.s[t1-1];
temps[t1-1]='0';
tempv=kangtuo(temps);
if(!vis1[tempv])
{
vis1[tempv]=1;
cur.loc=t1;
cur.d=dir1[i];
cur.fa=head1;
cur.step=now.step+1;
for(j=0; j<9; j++)
{
cur.s[j]=temps[j];
}
cur.val=tempv;
q1[++tail1]=cur;
mp1[tempv]=tail1;
}
}
head1++;
}
step1++;
while(q2[head2].step<=step2&&head2<=tail2)
{
now=q2[head2];
t=now.loc;
nval=now.val;
if(vis1[nval])
{
ans1=mp1[nval];
ans2=head2;
return true ;
}
y=t%3;
if(y==0) y=3;
x=(t-y)/3+1;
for(i=0; i<4; i++)
{
nx=x+dx[i];
ny=y+dy[i];
if(!(nx>=1&&nx<=3&&ny>=1&&ny<=3)) continue ;
t1=(nx-1)*3+ny;
for(j=0; j<9; j++)
{
temps[j]=now.s[j];
}
temps[t-1]=now.s[t1-1];
temps[t1-1]='0';
tempv=kangtuo(temps);
if(!vis2[tempv])
{
vis2[tempv]=1;
cur.loc=t1;
cur.d=dir2[i];
cur.fa=head2;
cur.step=now.step+1;
for(j=0; j<9; j++)
{
cur.s[j]=temps[j];
}
cur.val=tempv;
q2[++tail2]=cur;
mp2[tempv]=tail2;
}
}
head2++;
}
step2++;
}
}
void output1(int k) // 递归输出答案
{
if(k==0) return ;
else
{
output1(q1[k].fa);
printf("%c",q1[k].d);
}
}
void output2(int k) // 递归输出答案
{
if(k==0) return ;
else
{
printf("%c",q2[k].d);
output2(q2[k].fa);
}
}
int main()
{
int i,j,temp1,vv;
char c;
while(cin>>c)
{
if(c=='x')
{
cur.loc=1;
cur.s[0]='0';
}
else cur.s[0]=c;
for(i=1; i<9; i++)
{
cin>>c;
if(c=='x')
{
cur.loc=i+1;
cur.s[i]='0';
}
else cur.s[i]=c;
}
if(judge(cur.s)&&bfs())
{
output1(ans1);
output2(ans2);
printf("\n");
}
else printf("unsolvable\n");
}
return 0;
}
思路四：
IDA*，开始想的启发式函数：h()={ 不在自己位置上的个数 }，还是TLE，后来想想，这个不能怎么提高效率，因为从“不在自己位置上”到达“在自己位置上”所需的步数有差异，所以不能采用这个函数。考虑到每次移动实质上改变的是行或者列的坐标，可得到正确的h()={ 当前所在位置和本应该所在位置的行列之差的绝对值之和 }，另外还有一个剪枝见代码。
代码：
// 168K 0MS
#include <cstdio>
#include <cstring>
#include <cmath>
#define maxn 10
#define INF 0x3f3f3f3f
using namespace std;
int depth,flag,mi;
int a[maxn];
int dx[]= {-1,1,0,0}; // up down left right
int dy[]= {0,0,-1,1};
char dir[]="udlr";
char s[maxn],anss[1005];
bool judge() // 根据逆序数是否为奇数判断是否有解
{
int i,j,t=0;
for(i=0; i<9; i++)
{
if(a[i]==9) continue ;
for(j=i+1; j<9; j++)
{
if(a[i]>a[j]) t++;
}
}
if(t&1) return false ;
return true ;
}
int h() // 最少还需多少步达到目标状态
{
int i,j,nx,ny,tx,ty,t=0;
for(i=0; i<9; i++)
{
if(a[i]==9) continue ;
nx=i/3,ny=i%3;
tx=(a[i]-1)/3,ty=(a[i]-1)%3;
t=t+abs(nx-tx)+abs(ny-ty);
}
return t;
}
void dfs(int x,int d)
{
int i,j,t,nx,ny,tx,ty;
t=h();
if(mi>t) mi=t;
if(d+t>depth||flag) return ;
if(!t)
{
flag=1;
anss[d]='\0';
printf("%s\n",anss);
return ;
}
nx=x/3,ny=x%3;
for(i=0; i<4; i++)
{
tx=nx+dx[i];
ty=ny+dy[i];
if(tx<0||tx>=3||ty<0||ty>=3) continue ;
if((d>0)&&(i==0&&anss[d-1]=='d'||i==1&&anss[d-1]=='u'||i==2&&anss[d-1]=='r'||i==3&&anss[d-1]=='l')) continue ; // 剪枝 相邻两步不来回走
t=a[nx*3+ny],a[nx*3+ny]=a[tx*3+ty],a[tx*3+ty]=t;
anss[d]=dir[i];
dfs(tx*3+ty,d+1);
t=a[nx*3+ny],a[nx*3+ny]=a[tx*3+ty],a[tx*3+ty]=t;
}
}
void IDAX(int x)
{
int i,j;
depth=h();
flag=0;
while(!flag)
{
mi=INF;
dfs(x,0);
depth+=mi; // 每次加上最小需移动的步数(也可以++,稍稍慢一点)
}
}
int main()
{
int i,j,x;
while(~scanf("%s",s))
{
if(s[0]=='x') x=0,a[0]=9;
else a[0]=s[0]-'0';
for(i=1; i<=8; i++)
{
scanf("%s",s);
if(s[0]=='x') x=i,a[i]=9;
else a[i]=s[0]-'0';
}
if(!judge())
{
printf("unsolvable\n");
continue ;
}
IDAX(x);
}
return 0;
}
思路五：
A*，启发式函数和上面的IDA*一样的，用优先队列实现。（代码在原来单项bfs基础上改的，有点囧）
代码：
// 912K 32MS
#include <iostream>
#include <cstdio>
#include <cstring>
#include <string>
#include <cmath>
#include <queue>
#define maxn 370000
using namespace std;
int ans,sx;
int fac[]= {40320,5040,720,120,24,6,2,1,1}; //康拖展开判重
bool vis[maxn];
int dx[]= {-1,1,0,0}; // up down left right
int dy[]= {0,0,-1,1};
char dir[]="udlr";
struct Node
{
char d;
int fa;
}anss[maxn];
struct Tnode
{
char s[9];
int loc; // “0”的位置
int val; // 康拖展开的值
int hh; // 到达目标状态最小步数
bool operator < (const Tnode&xx)const
{
return hh>xx.hh;
}
} cur,now;
priority_queue<Tnode>q;
bool judge(char *ss) // 根据逆序数是否为奇数判断是否有解
{
int i,j,t=0;
for(i=0; i<9; i++)
{
if(ss[i]=='0') continue ;
for(j=i+1; j<9; j++)
{
if(ss[i]>ss[j]&&ss[j]!='0') t++;
}
}
if(t&1) return false ;
return true ;
}
int kangtuo(char *ss) // 求康拓展开的值
{
int i,j,t,sum=0;
for(i=0; i<9; i++)
{
t=0;
for(j=i+1; j<9; j++)
{
if(ss[i]>ss[j]) t++;
}
sum+=t*fac[i];
}
return sum+1;
}
int h(char *ss) // 最少还需多少步达到目标状态
{
int i,j,nx,ny,tx,ty,t=0;
for(i=0; i<9; i++)
{
if(ss[i]=='0') continue ;
nx=i/3,ny=i%3;
tx=(ss[i]-'1')/3,ty=(ss[i]-'1')%3;
t=t+abs(nx-tx)+abs(ny-ty);
}
return t;
}
bool bfs()
{
int i,j,t,t1;
int tempv,x,y,nx,ny,nval,npos;
char temps[9];
memset(vis,0,sizeof(vis));
while(!q.empty()) q.pop();
cur.hh=h(cur.s);
cur.val=kangtuo(cur.s);
sx=cur.val;
vis[cur.val]=1;
q.push(cur);
while(!q.empty())
{
now=q.top();
q.pop();
t=now.loc;
nval=now.val;
y=t%3;
if(y==0) y=3;
x=(t-y)/3+1;
for(i=0; i<4; i++)
{
nx=x+dx[i];
ny=y+dy[i];
if(!(nx>=1&&nx<=3&&ny>=1&&ny<=3)) continue ;
t1=(nx-1)*3+ny;
for(j=0; j<9; j++)
{
temps[j]=now.s[j];
}
temps[t-1]=now.s[t1-1];
temps[t1-1]='0';
tempv=kangtuo(temps);
if(!vis[tempv])
{
if(tempv==46234)
{
anss[tempv].d=dir[i];
anss[tempv].fa=nval;
return true ;
}
vis[tempv]=1;
cur.loc=t1;
anss[tempv].d=dir[i];
anss[tempv].fa=nval;
for(j=0; j<9; j++)
{
cur.s[j]=temps[j];
}
cur.hh=h(cur.s);
cur.val=tempv;
q.push(cur);
}
}
}
return false ;
}
void output(int k) // 递归输出答案
{
if(k==sx) return ;
else
{
output(anss[k].fa);
printf("%c",anss[k].d);
}
}
int main()
{
int i,j,temp1,vv;
char c;
ans=46234;
while(cin>>c)
{
if(c=='x')
{
cur.loc=1;
cur.s[0]='0';
}
else cur.s[0]=c;
for(i=1; i<9; i++)
{
cin>>c;
if(c=='x')
{
cur.loc=i+1;
cur.s[i]='0';
}
else cur.s[i]=c;
}
if(judge(cur.s)&&bfs())
{
output(ans);
printf("\n");
}
else printf("unsolvable\n");
}
return 0;
}