2017 ACM-ICPC 亞洲區（南寧賽區）網絡賽 A Weather Patterns

Consider a system which is described at any time as being in one of a set of $N$ distinct states, $1,2,3,...,N$. We denote the time instants associated with state changes as $t=1,2,...$, and the actual state at time $t$ as a_{ij}=p=[s_{i}=j\ |\ s_{i-1}=i], 1\le i,j \le Na​ij​​=p=[s​i​​=j ∣ s​i−1​​=i],1≤i,j≤N. For the special case of a discrete, first order, Markovchain, the probabilistic description for the current state (at time $t$) and the predecessor state is s_{t}s​t​​. Furthermore we only consider those processes being independent of time, thereby leading to the set of state transition probability a_{ij}a​ij​​ of the form: with the properties a_{ij} \geq 0a​ij​​≥0 and \sum_{i=1}^{N} A_{ij} = 1∑​i=1​N​​A​ij​​=1. The stochastic process can be called an observable Markovmodel. Now, let us consider the problem of a simple 4-state Markov model of weather. We assume that once a day (e.g., at noon), the weather is observed as being one of the following:

State $1$: snow

State $2$: rain

State $3$: cloudy

State $4$: sunny

The matrix A of state transition probabilities is:

A = \{a_{ij}\}= \begin{Bmatrix} a_{11}&a_{12}&a_{13}&a_{14} \\ a_{21}&a_{22}&a_{23}&a_{24} \\ a_{31}&a_{32}&a_{33}&a_{34} \\ a_{41}&a_{42}&a_{43}&a_{44} \end{Bmatrix}A={a​ij​​}=​⎩​⎪​⎪​⎨​⎪​⎪​⎧​​​a​11​​​a​21​​​a​31​​​a​41​​​​​a​12​​​a​22​​​a​32​​​a​42​​​​​a​13​​​a​23​​​a​33​​​a​43​​​​​a​14​​​a​24​​​a​34​​​a​44​​​​​⎭​⎪​⎪​⎬​⎪​⎪​⎫​​

Given the model, several interestingquestions about weather patterns over time can be asked (and answered). We canask the question: what is the probability (according to the given model) thatthe weather for the next k days willbe? Another interesting question we can ask: given that the model is in a knownstate, what is the expected number of consecutive days to stay in that state?Let us define the observation sequence $O$as O = \left \{ s_{1}, s_{2}, s_{3}, ... , s_{k} \right \}O={s​1​​,s​2​​,s​3​​,...,s​k​​}, and the probability of the observation sequence $O$given the model is defined as $p(O|model)$. Also, let the expected number of consecutive days to stayin state $i$ be E_{i}E​i​​. Assume that the initial state probabilities p[s_{1} = i] = 1, 1 \leq i \leq Np[s​1​​=i]=1,1≤i≤N. Both$p(O|model)$and E_{i}E​i​​ are real numbers.

Input Format

Line $1$~$4$ for the state transition probabilities. Line $5$ for the observation sequence O_{1}O​1​​, and line $6$ for the observation sequence O_{2}O​2​​. Line $7$and line $8$ for the states of interest to find the expected number of consecutive days to stay in these states.

Line $1$: a_{11}\ a_{12}\ a_{13}\ a_{14}a​11​​ a​12​​ a​13​​ a​14​​

Line $2$: a_{21}\ a_{22}\ a_{23}\ a_{34}a​21​​ a​22​​ a​23​​ a​34​​

Line $3$: a_{31}\ a_{32}\ a_{33}\ a_{34}a​31​​ a​32​​ a​33​​ a​34​​

Line $4$: a_{41}\ a_{42}\ a_{43}\ a_{44}a​41​​ a​42​​ a​43​​ a​44​​

Line $5$: s_{1}\ s_{2}\ s_{3}\ ...\ s_{k}s​1​​ s​2​​ s​3​​ ... s​k​​

Line $6$: s_{1}\ s_{2}\ s_{3}\ ...\ s_{l}s​1​​ s​2​​ s​3​​ ... s​l​​

Line $7$: $i$

Line $8$: $j$

Output Format

Line $1$ and line $2$ are used to show the probabilities of the observation sequences O_{1}O​1​​and O_{2}O​2​​ respectively. Line $3$ and line $4$ are for the expected number of consecutive days to stay in states $i$ and $j$ respectively.

Line $1$: p[O_{1} | model]p[O​1​​∣model]

Line $2$: p[O_{2} | model]p[O​2​​∣model]

Line $3$: E_{i}E​i​​

Line $4$: E_{j}E​j​​

Please be reminded that the floating number should accurate to 10^{-8}10​−8​​.

樣例輸入

0.4 0.3 0.2 0.1
0.3 0.3 0.3 0.1
0.1 0.1 0.6 0.2
0.1 0.2 0.2 0.5
4 4 3 2 2 1 1 3 3
2 1 1 1 3 3 4
3
4

樣例輸出

0.00004320
0.00115200
2.50000000
2.00000000

題目來源

2017 ACM-ICPC 亞洲區（南寧賽區）網絡賽

題意：先給你4 * 4 的矩陣，表示天氣 i 到天氣 j 轉變的概率，然後輸入 k 個數字，以 ‘\n’ 表示輸入結束，這 k 個數字的意思以

4 4 3 2 2 1 1 3 3

這組數字舉例，其實就是天氣從 4 轉變爲 4，然後從 4 轉變爲 3，然後從 3 轉變爲 2，然後從 2 轉變爲 2，然後從 2 轉變爲 1，然後從 1 轉變爲 1，然後從1 轉變爲 3，然後從3 轉變爲 3的概率是多少。

這樣的輸入數據有兩組

然後輸入兩個數字 i 和 j ，這兩個數字都表示天氣的狀況爲 i 和 j 一直不變，不管天數增加多少，然後要你求一直不變的數學期望是多少。

附代碼：

#include <cstdio>
#include <cstring>
#include <algorithm>
#include <iostream>
#include <cmath>
#include <vector>
#define mem(a, b) memset(a, b, sizeof(a))
using namespace std;
typedef long long LL;
///o(っ。Д。)っ AC萬歲!!!!!!!!!!!!!!
const int maxn = 10;
double maps[maxn][maxn] = {};


int main()
{
    for(int i = 1; i <= 4; i++)
    {
        for(int j = 1; j <= 4; j++)
        {
            scanf("%lf", &maps[i][j]);
        }
    }
    double pp = 1.0;
    int d1, d2;
    char ch;
    scanf("%d", &d1);
    while(scanf("%d%c", &d2, &ch) != EOF)
    {
        pp *= maps[d1][d2];
        d1 = d2;
        if(ch == '\n') break;
    }
    printf("%.8lf\n", pp);
    pp = 1.0;
    scanf("%d", &d1);
    while(scanf("%d%c", &d2, &ch) != EOF)
    {
        pp *= maps[d1][d2];
        d1 = d2;
        if(ch == '\n') break;
    }
    printf("%.8lf\n", pp);
    scanf("%d", &d1);
    pp = maps[d1][d1];
    double sum = 0.0;
    while(pp >= 0.000000001)
    {
        sum += pp;
        pp *= maps[d1][d1];
    }
    printf("%.8lf\n", sum + 1.0);
    sum = 0.0;
    scanf("%d", &d2);
    pp = maps[d2][d2];
    while(pp >= 0.000000001)
    {
        sum += pp;
        pp *= maps[d2][d2];
    }
    printf("%.8lf\n", sum + 1.0);
    return 0;
}
/*


*/