hdu 4405 Aeroplane chess （概率dp）

最新推荐文章于 2021-03-30 14:32:48 发布

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本文探讨了飞机棋游戏中的期望掷骰子次数的计算方法，包括游戏规则、状态转移矩阵的构建以及动态规划算法的应用。通过实例分析，详细解释了如何计算从起点到终点的期望步数，提供了具体的数学模型和编程实现。

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Aeroplane chess

Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 32768/32768 K (Java/Others)

Total Submission(s): 952 Accepted Submission(s): 652

Problem Description

Hzz loves aeroplane chess very much. The chess map contains N+1 grids labeled from 0 to N. Hzz starts at grid 0. For each step he throws a dice(a dice have six faces with equal probability to face up and the numbers on the faces are 1,2,3,4,5,6). When Hzz is at grid i and the dice number is x, he will moves to grid i+x. Hzz finishes the game when i+x is equal to or greater than N.

There are also M flight lines on the chess map. The i-th flight line can help Hzz fly from grid Xi to Yi (0<Xi<Yi<=N) without throwing the dice. If there is another flight line from Yi, Hzz can take the flight line continuously. It is granted that there is no two or more flight lines start from the same grid.

Please help Hzz calculate the expected dice throwing times to finish the game.

Input

There are multiple test cases.
Each test case contains several lines.
The first line contains two integers N(1≤N≤100000) and M(0≤M≤1000).
Then M lines follow, each line contains two integers Xi,Yi(1≤Xi<Yi≤N).
The input end with N=0, M=0.

Output

For each test case in the input, you should output a line indicating the expected dice throwing times. Output should be rounded to 4 digits after decimal point.

Sample Input

Sample Output

  
   1.1667
2.3441

Source

2012 ACM/ICPC Asia Regional Jinhua Online

思路：

dp[i]表示i到目标位置的期望，所以dp[i]=∑dp[j]/6+1 (i<j<=i+6)，不过i->j的话当然就是dp[i]=dp[j]喽。

代码：

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <cmath>
#include <string>
#include <map>
#include <stack>
#include <vector>
#include <set>
#include <queue>
#pragma comment (linker,"/STACK:102400000,102400000")
#define maxn 105
#define MAXN 100025
#define mod 100000007
#define INF 0x3f3f3f3f
#define pi acos(-1.0)
#define eps 1e-6
typedef long long ll;
using namespace std;

int n,m;
int mp[MAXN];
double ans;
double dp[MAXN];

void solve()
{
    int i,j;
    memset(dp,0,sizeof(dp));
    for(i=n-1;i>=0;i--)
    {
        if(mp[i])
        {
            dp[i]=dp[mp[i]];
            continue ;
        }
        for(j=1;j<=6;j++)
        {
            dp[i]=dp[i]+dp[i+j]/6.0;
        }
        dp[i]++;
    }
    ans=dp[0];
}
int main()
{
    int i,j,t,x,y;
    while(scanf("%d%d",&n,&m),n||m)
    {
        memset(mp,0,sizeof(mp));
        for(i=1;i<=m;i++)
        {
            scanf("%d%d",&x,&y);
            mp[x]=y;
        }
        solve();
        printf("%.4f\n",ans);
    }
    return 0;
}