HDU 1043 & POJ 1077 Eight（康托展开+BFS | IDA*）

Eight

Time Limit: 1000MS		Memory Limit: 65536K
Total Submissions: 30176		Accepted: 13119		Special Judge

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 a description of a configuration of the 8 puzzle. The 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.

Sample Input

 2  3  4  1  5  x  7  6  8 

Sample Output

ullddrurdllurdruldr

Source

South Central USA 1998

题目链接：POJ 1077

主要就是用康托展开来映射判重的问题，info::val就是康托展开hash值，info::step就是积累的状态。另外感觉这题剧毒，自己本来用int vis[]和char his[]想最后回溯记录答案从而代替速度比较慢的string，结果居然超时……TLE一晚上，要不是看了大牛的博客估计要一直T在这个坑点上。还有不知道为什么string的加号重载在C++编译器里会CE，换G++才过。相比单组输入的POJ，多组输入的HDU就友好多了，打个表就可以水过了，双广、A*神马的写起来麻烦就先不写了……

什么是康托展开？——康托展开介绍文章

POJ代码：

#include<iostream>
#include<algorithm>
#include<cstdlib>
#include<sstream>
#include<cstring>
#include<bitset>
#include<cstdio>
#include<string>
#include<deque>
#include<stack>
#include<cmath>
#include<queue>
#include<set>
#include<map>
using namespace std;
#define INF 0x3f3f3f3f
#define CLR(x,y) memset(x,y,sizeof(x))
#define LC(x) (x<<1)
#define RC(x) ((x<<1)+1)
#define MID(x,y) ((x+y)>>1)
typedef pair<int, int> pii;
typedef long long LL;
const double PI = acos(-1.0);
const int N = 362880 + 20;
int fact[10] = {1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880};
int direct[4][2] = {{ -1, 0}, {1, 0}, {0, -1}, {0, 1}};
char MOVE[5] = "udlr";
struct info
{
    int s[9];
    int indx;
    string step;
    int val;
};
info S, E;
int T;
int vis[N];
string ans;
int calcantor(int s[])
{
    int r = 0;
    for (int i = 0; i < 9; ++i)
    {
        int k = 0;
        for (int j = i + 1; j < 9; ++j)
        {
            if (s[j] < s[i])
                ++k;
        }
        r = r + k * fact[8 - i];
    }
    return r;
}
bool check(const int &x, const int &y)
{
    return (x >= 0 && x < 3 && y >= 0 && y < 3);
}
bool bfs()
{
    CLR(vis, 0);
    queue<info>Q;
    Q.push(S);
    vis[S.val] = 1;
    info now, v;
    while (!Q.empty())
    {
        now = Q.front();
        Q.pop();
        if (now.val == T)
        {
            ans = now.step;
            return true;
        }
        for (int i = 0; i < 4; ++i)
        {
            int x = now.indx / 3;
            int y = now.indx % 3;
            x += direct[i][0];
            y += direct[i][1];
            if (check(x, y))
            {
                v = now;
                v.indx = x * 3 + y;
                v.s[now.indx] = v.s[v.indx];
                v.s[v.indx] = 0;
                v.val = calcantor(v.s);
                if (!vis[v.val])
                {
                    vis[v.val] = 1;
                    v.step = now.step + MOVE[i];
                    if (v.val == T)
                    {
                        ans = v.step;
                        return true;
                    }
                    Q.push(v);
                }
            }
        }
    }
    return false;
}
int main(void)
{
    char temp;
    int i;
    T = 46233;
    while (cin >> temp)
    {
        if (temp == 'x')
        {
            S.s[0] = 0;
            S.indx = 0;
        }
        else
            S.s[0] = temp - '0';
        for (i = 1; i < 9; ++i)
        {
            cin >> temp;
            if (temp == 'x')
            {
                S.s[i] = 0;
                S.indx = i;
            }
            else
                S.s[i] = temp - '0';
        }
        S.val = calcantor(S.s);
        puts(!bfs() ? "unsolvable" : ans.c_str());
    }
    return 0;
}

HDU代码：

#include<iostream>
#include<algorithm>
#include<cstdlib>
#include<sstream>
#include<cstring>
#include<bitset>
#include<cstdio>
#include<string>
#include<deque>
#include<stack>
#include<cmath>
#include<queue>
#include<set>
#include<map>
using namespace std;
#define INF 0x3f3f3f3f
#define CLR(x,y) memset(x,y,sizeof(x))
#define LC(x) (x<<1)
#define RC(x) ((x<<1)+1)
#define MID(x,y) ((x+y)>>1)
typedef pair<int, int> pii;
typedef long long LL;
const double PI = acos(-1.0);
const int N = 362880 + 10;
int fact[10] = {1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880};
struct info
{
    int s[9];
    int indx;
    string path;
    int val;
    void cal()
    {
        val = 0;
        for (int i = 0; i < 9; ++i)
        {
            int k = 0;
            for (int j = i + 1; j < 9; ++j)
            {
                if (s[j] < s[i])
                    ++k;
            }
            val = val + k * fact[8 - i];
        }
    }
};
info init = {{1, 2, 3, 4, 5, 6, 7, 8, 9}, 8, "", 0}, S;
int direct[4][2] = {{ -1, 0}, {1, 0}, {0, -1}, {0, 1}}; //ÏÂÉÏÓÒ×ó
char MOV[5] = "durl";
string ans[N];
int vis[N];
void bfs()
{
    CLR(vis, 0);
    queue<info>Q;
    info now, v;
    Q.push(init);
    vis[init.val] = 1;
    ans[init.val] = init.path;
    while (!Q.empty())
    {
        now = Q.front();
        Q.pop();
        for (int i = 0; i < 4; ++i) //ÏÂÉÏÓÒ×ó
        {
            int x = now.indx / 3;
            int y = now.indx % 3;
            x += direct[i][0];
            y += direct[i][1];
            if (x >= 0 && x < 3 && y >= 0 && y < 3)
            {
                v = now;
                v.indx = x * 3 + y;
                v.path = MOV[i] + now.path;
                swap(v.s[v.indx], v.s[now.indx]);
                v.cal();
                if (!vis[v.val])
                {
                    vis[v.val] = 1;
                    ans[v.val] = v.path;
                    Q.push(v);
                }
            }
        }
    }
}
int main(void)
{
    char temp[5], i;
    bfs();
    while (~scanf("%s", temp))
    {
        S.path = "";
        if (temp[0] == 'x')
        {
            S.s[0] = 9;
            S.indx = 0;
        }
        else
            S.s[0] = temp[0] - '0';
        for (i = 1; i < 9; ++i)
        {
            scanf("%s", temp);
            if (temp[0] == 'x')
            {
                S.s[i] = 9;
                S.indx = i;
            }
            else
                S.s[i] = temp[0] - '0';
        }
        S.cal();
        puts(!vis[S.val] ? "unsolvable" : ans[S.val].c_str());
    }
    return 0;
}

---

最近学了下IDA*，发现速度贼快，比哈希的不知道高到哪里去了，自己整理了一下一般写法，感觉还是比较模版的，可以参考上一篇IDA*的伪代码，IDA*快到如果用BFS要打表的HDU上的数据可以直接在线搜索过了，确实比较快，这里需要加一个防止走回路的剪枝（有效减少无用搜索），否则可能会超时，当然一开始要判断一下是否输入序列是无解的，可以暂时把x忽略掉，然后算7个数的逆序数，逆序数偶数才有解，奇数直接输出unsolvable，具体原理可以百度一下

代码：

#include <stdio.h>
#include <iostream>
#include <algorithm>
#include <cstdlib>
#include <sstream>
#include <numeric>
#include <cstring>
#include <bitset>
#include <string>
#include <deque>
#include <stack>
#include <cmath>
#include <queue>
#include <set>
#include <map>
using namespace std;
#define INF 0x3f3f3f3f
#define LC(x) (x<<1)
#define RC(x) ((x<<1)+1)
#define MID(x,y) ((x+y)>>1)
#define CLR(arr,val) memset(arr,val,sizeof(arr))
#define FAST_IO ios::sync_with_stdio(false);cin.tie(0);
typedef pair<int, int> pii;
typedef long long LL;
const double PI = acos(-1.0);
const int N = 9;
int arr[N], T[N] = {1, 2, 3, 4, 5, 6, 7, 8, 0};
int Maxdep, top, stdpos[N] = {8, 0, 1, 2, 3, 4, 5, 6, 7};
char Ops[1010];

int dis(int cur, int ori)
{
    int x = cur / 3, y = cur % 3;
    return abs(x - ori / 3) + abs(y - ori % 3);
}
int Need(int A[])
{
    int ret = 0;
    for (int i = 0; i < N; ++i)
        if (A[i] != T[i])
            ret += dis(i, stdpos[A[i]]);
    return ret;
}
inline bool check(int x, int y)
{
    return x >= 0 && x < 3 && y >= 0 && y < 3;
}
int dfs(int dep, int Arr[], char pre)
{
    int need = Need(Arr);
    if (need == 0)
        return 1;
    else if (need + dep > Maxdep)
        return 0;
    else
    {
        int x, y, idx, i;
        int Temp[N];
        for (int i = 0; i < N; ++i)
        {
            if (Arr[i] == 0)
            {
                idx = i;
                break;
            }
        }
        x = idx / 3, y = idx % 3;
        if (pre != 'u' && check(x + 1, y))
        {
            for (i = 0; i < N; ++i)
                Temp[i] = Arr[i];
            swap(Temp[(x + 1) * 3 + y], Temp[idx]);
            Ops[top++] = 'd';
            if (dfs(dep + 1, Temp, 'd'))
                return 1;
            else
                --top;
        }
        if (pre != 'l' && check(x, y + 1))
        {
            for (i = 0; i < N; ++i)
                Temp[i] = Arr[i];
            swap(Temp[x * 3 + y + 1], Temp[idx]);
            Ops[top++] = 'r';
            if (dfs(dep + 1, Temp, 'r'))
                return 1;
            else
                --top;
        }
        if (pre != 'd' && check(x - 1, y))
        {
            for (i = 0; i < N; ++i)
                Temp[i] = Arr[i];
            swap(Temp[(x - 1) * 3 + y], Temp[idx]);
            Ops[top++] = 'u';
            if (dfs(dep + 1, Temp, 'u'))
                return 1;
            else
                --top;
        }
        if (pre != 'r' && check(x, y - 1))
        {
            for (i = 0; i < N; ++i)
                Temp[i] = Arr[i];
            swap(Temp[x * 3 + y - 1], Temp[idx]);
            Ops[top++] = 'l';
            if (dfs(dep + 1, Temp, 'l'))
                return 1;
            else
                --top;
        }
    }
    return 0;
}
int main(void)
{
    char n[3];
    int i, j, x;
    while (~scanf("%s", n))
    {
        if (n[0] == 'x')
            x = 0;
        else
            x = n[0] - '0';
        arr[0] = x;
        for (i = 1; i < 9; ++i)
        {
            scanf("%s", n);
            if (n[0] == 'x')
                x = 0;
            else
                x = n[0] - '0';
            arr[i] = x;
        }
        int inv = 0;
        for (i = 0; i < N; ++i)
        {
            if (arr[i] == 0)
                continue;
            for (j = i + 1; j < N; ++j)
            {
                if (arr[j] == 0)
                    continue;
                if (arr[i] > arr[j])
                    ++inv;
            }
        }
        if (inv & 1)
            puts("unsolvable");
        else
        {
            top = 0;
            Maxdep = 1;
            while (!dfs(0, arr, -1))
                ++Maxdep;
            Ops[top] = '\0';
            puts(Ops);
        }
    }
    return 0;
}