ZOJ Problem Set - 3329(概率DP)

One Person Game

There is a very simple and interesting one-person game. You have 3 dice, namely Die1, Die2 and Die3. Die1 has K₁ faces. Die2 has K₂ faces. Die3 has K₃ faces. All the dice are fair dice, so the probability of rolling each value, 1 to K₁, K₂, K₃ is exactly 1 / K₁, 1 / K₂ and 1 / K₃. You have a counter, and the game is played as follow:

1. Set the counter to 0 at first.
2. Roll the 3 dice simultaneously. If the up-facing number of Die1 is a, the up-facing number of Die2 is b and the up-facing number of Die3 is c, set the counter to 0. Otherwise, add the counter by the total value of the 3 up-facing numbers.
3. If the counter‘s number is still not greater than n, go to step 2. Otherwise the game is ended.

Calculate the expectation of the number of times that you cast dice before the end of the game.

Input

There are multiple test cases. The first line of input is an integer T (0 < T <= 300) indicating the number of test cases. Then T test cases follow. Each test case is a line contains 7 non-negative integers n, K₁, K₂, K₃, a, b, c (0 <= n <= 500, 1 < K₁, K₂, K₃ <= 6, 1 <= a <= K₁, 1 <= b <= K₂, 1 <= c <= K₃).

Output

For each test case, output the answer in a single line. A relative error of 1e-8 will be accepted.

Sample Input

2
0 2 2 2 1 1 1
0 6 6 6 1 1 1

Sample Output

1.142857142857143
1.004651162790698

本题通过代换系数，化简后求系数。

一般形成环的用高斯消元法求解。但是此题都是和dp[0]相关。所有可以分离出系数。

dp[i]表示达到i还要掷几次的期望，每一项都和dp[0]有关，且可表示成dp[i]=A[i]*dp[0]+B[0];

所以只要求出dp[0]的系数A,B就可以求出dp[0]=B[0]/(1-A[0]);

dp[n] = dp[0]/k1/k1/k1+1;

然后递推可推出dp[0]的系数;

 1 #include<iostream>
 2 #include<cstdio>
 3 #include<cstring>
 4 #define M(a,b) memset(a,b,sizeof(a))
 5
 6 using namespace std;
 7
 8 double A[1005],B[1005];
 9 int n,k1,k2,k3,a,b,c;
10
11 int main()
12 {
13     int t;
14     scanf("%d",&t);
15     while(t--)
16     {
17         M(A,0);
18         M(B,0);
19         scanf("%d%d%d%d%d%d%d",&n,&k1,&k2,&k3,&a,&b,&c);
20         A[n] = 1.0/(k1*k2*k3);
21         B[n] = 1;
22         for(int i = n-1;i>=0;i--)
23         {
24             for(int p = 1;p<=k1;p++)
25                 for(int q = 1;q<=k2;q++)
26                    for(int r = 1;r<=k3;r++)
27             {
28                 if(p!=a||q!=b||r!=c)
29                    {
30                        A[i] += A[i+p+q+r]/(k1*k2*k3);
31                        B[i] += B[i+p+q+r]/(k1*k2*k3);
32                    }
33                    //cout<<A[i]<<‘ ‘<<B[i]<<endl;
34             }
35             A[i]+=(1.0/(k1*k2*k3));
36             B[i]+=1;
37             //cout<<A[i]<<‘ ‘<<B[i]<<endl;
38         }
39         double ans = B[0]/(1-A[0]);
40         printf("%.16f\n",ans);
41     }
42     return 0;
43 }