Scout YYF I_poj3744

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

题意：在一条不满地雷的路上，你现在的起点在1处。在N个点处布有地雷，1<=N<=10。地雷点的坐标范围：[1,100000000].

每次前进p的概率前进一步，1-p的概率前进1-p步。问顺利通过这条路的概率。就是不要走到有地雷的地方。

 

设dp[i]表示到达i点的概率，则 初始值 dp[1]=1.

很容易想到转移方程： dp[i]=p*dp[i-1]+(1-p)*dp[i-2];

但是由于坐标的范围很大，直接这样求是不行的，而且当中的某些点还存在地雷。

 

N个有地雷的点的坐标为 x[1],x[2],x[3]```````x[N].

我们把道路分成N段：

1~x[1];

x[1]+1~x[2];

x[2]+1~x[3];

`

`

`

x[N-1]+1~x[N].

 

这样每一段只有一个地雷。我们只要求得通过每一段的概率。乘法原理相乘就是答案。

对于每一段，通过该段的概率等于1-踩到该段终点的地雷的概率。

 

就比如第一段 1~x[1].  通过该段其实就相当于是到达x[1]+1点。那么p[x[1]+1]=1-p[x[1]].

但是这个前提是p[1]=1,即起点的概率等于1.对于后面的段我们也是一样的假设，这样就乘起来就是答案了。

 

对于每一段的概率的求法可以通过矩阵乘法快速求出来。

View Code

  1 #include<cstdio>

  2 

  3 #include<cstring>

  4 

  5 #include<algorithm>

  6 

  7 using namespace std;

  8 

  9 struct Mat

 10 

 11 {

 12 

 13     double d[2][2];

 14 

 15 };

 16 

 17 #define maxn 30

 18 

 19 int x[maxn];

 20 

 21 Mat mul(Mat a,Mat b)

 22 

 23 {

 24 

 25     Mat temp;

 26 

 27     for(int i=0;i<2;i++)

 28 

 29         for(int j=0;j<2;j++)

 30 

 31         {

 32 

 33             temp.d[i][j]=0;

 34 

 35             for(int k=0;k<2;k++)

 36 

 37                 temp.d[i][j]+= (a.d[i][k]* b.d[k][j]);

 38 

 39         }

 40 

 41     return temp;

 42 

 43 }

 44 

 45 Mat pow(Mat a,int n)

 46 

 47 {

 48 

 49     Mat temp;

 50 

 51     temp=a;

 52 

 53     Mat ret;

 54 

 55     ret.d[0][0]=1.0;

 56 

 57     ret.d[0][1]=0.0;

 58 

 59     ret.d[1][0]=0.0;

 60 

 61     ret.d[1][1]=1.0;

 62 

 63     while(n)

 64 

 65     {

 66 

 67         if(n&1)

 68 

 69             ret=mul(ret,temp);

 70 

 71         temp=mul(temp,temp);

 72 

 73         n>>=1;

 74 

 75     }

 76 

 77     return ret;

 78 

 79 }

 80 

 81 int main()

 82 

 83 {

 84 

 85     int n;double p;

 86 

 87     while(scanf("%d%lf",&n,&p)!=EOF)

 88 

 89     {

 90 

 91         Mat tt;

 92 

 93         tt.d[0][0]=p;

 94 

 95         tt.d[0][1]=1-p;

 96 

 97         tt.d[1][0]=1;

 98 

 99         tt.d[1][1]=0;

100 

101         for(int i=0;i<n;i++)

102 

103             scanf("%d",&x[i]);

104 

105         sort(x,x+n);

106 

107         double ans=1;

108 

109         Mat temp;

110 

111         temp=pow(tt,x[0]-1);

112 

113         ans*=(1-temp.d[0][0]);

114 

115  

116 

117         for(int i=1;i<n;i++)

118 

119         {

120 

121             if(x[i]==x[i-1])

122 

123                 continue;

124 

125             temp=pow(tt,x[i]-x[i-1]-1);

126 

127             ans*=(1-temp.d[0][0]);

128 

129         }

130 

131         printf("%.7f\n",ans);

132 

133     }

134 

135     return 0;

136 

137 }