POJ 3744|Scout YYF I|概率DP

令$dp[i]$表示到i的概率
显然有

$$dp[i]=p\times dp[i-1]+(1-p)\times dp[i-1]$$
然后状态转移很大，考虑矩阵优化。
显然有
$$\begin{bmatrix} dp[i-1]&dp[i-2] \end{bmatrix} \begin{bmatrix} p&1\\1-p&0 \end{bmatrix}= \begin{bmatrix} dp[i]&dp[i-1] \end{bmatrix}$$
如果有2个相邻的地雷，那么就不可能再继续走了。

#include <cstdio>
#include <algorithm>
using namespace std;
#define rep(i,j,k) for(i=j;i<k;++i)
#define FOR(i,j,k) for(i=j;i<=k;++i)
struct Matrix {
    double c[2][2];
    friend Matrix operator* (const Matrix &a, const Matrix &b) {
        Matrix c = (Matrix) { 0, 0, 0, 0 }; int i, j, k;
        rep(i,0,2) rep(j,0,2) rep(k,0,2) c.c[i][j] += a.c[i][k] * b.c[k][j];
        return c;
    }
    friend Matrix operator^ (Matrix a, int b) {
        Matrix c = (Matrix) { 1, 0, 1, 0 };
        for (; b; b >>= 1, a = a * a)
            if (b & 1) c = c * a;
        return c;
    }
};
int a[16];
int main() {
    int n, i, k; Matrix x, r; double p, ans;
    while (scanf("%d%lf", &n, &p) != EOF) {
        FOR(i,1,n) scanf("%d", a + i);
        sort(a + 1, a + n + 1);
        x = (Matrix) { p, 1, 1 - p, 0 };
        if (a[1] == 1) { printf("%.7f\n", 0.0); continue; }
        ans = 1; k = n + 1;
        r = (Matrix) { 1, 0, 1, 0 };
        FOR(i,1,n) {
            if (a[i] == a[i - 1]) continue;
            if (a[i] == a[i - 1] + 1) { k = i; break; }
            r = x ^ (a[i] - a[i - 1] - 2);
            ans *= r.c[0][0] * (1 - p);
        }
        printf("%.7f\n", k <= n ? 0.0 : ans);
    }
    return 0;
}

Scout YYF I

Time Limit: 1000MS

Description

YYF is a couragous scout. Now he is on a dangerous mission which is to penetrate into the enemy’s base. After overcoming a series difficulties, YYF is now at the start of enemy’s famous “mine road”. This is a very long road, on which there are numbers of mines. At first, YYF is at step one. For each step after that, YYF will walk one step with a probability of p, or jump two step with a probality of 1-p. Here is the task, given the place of each mine, please calculate the probality that YYF can go through the “mine road” safely.

Input

The input contains many test cases ended with EOF.
Each test case contains two lines.
The First line of each test case is N (1 ≤ N ≤ 10) and p (0.25 ≤ p ≤ 0.75) seperated by a single blank, standing for the number of mines and the probability to walk one step.
The Second line of each test case is N integer standing for the place of N mines. Each integer is in the range of [1, 100000000].

Output

For each test case, output the probabilty in a single line with the precision to 7 digits after the decimal point.

Sample Input

1 0.5
2
2 0.5
2 4

Sample Output

0.5000000
0.2500000