[测试题]圆圈

Description

现在有一个圆圈,顺时针标号分别从 0 到 n-1,每次等概率顺时针走一步或者逆时针走一步,即如果你在 i 号点,你有 1/2 概率走到((i-1)mod n)号点,1/2 概率走到((i+1)mod n)号点。问从 0 号点走到 x 号点的期望步数。

Input

第一行包含一个整数 T,表示数据的组数。
接下来有 T 行,每行包含两个整数 n, x。
T<=10, 0<=x<n<=300;​​,a​2​​⋯a​n​​ 和 bbb，代表一组方程。

Output

对于每组数据,输出一行,包含一个四位小数表示答案。

Sample Input

3
3 2
5 4
10 5

Sample Output

2.0000
4.0000
25.0000

HINT

对于 30% 的数据,n<=20
对于 50% 的数据,n<=100
对于 70% 的数据,n<=200
对于 100%的数据,n<=300

题解

高斯消元,$n$个方程，$n$个未知量, 设$E[ p ]$ 为从$p$节点走到$x$节点还需要走的步数的期望数。那么$E [ x ] = 0$;

对于每个节点都有   $E[p]=0.5*E[p-1]+0.5*E[p+1]+1$,  即  $-0.5*E[p-1]+E[p]-0.5*E[p+1]=1$。

模板套上一题的就好了。

//It is made by Awson on 2017.10.10
#include <set>
#include <map>
#include <cmath>
#include <ctime>
#include <cmath>
#include <stack>
#include <queue>
#include <vector>
#include <string>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <iostream>
#include <algorithm>
#define LL long long
#define Min(a, b) ((a) < (b) ? (a) : (b))
#define Max(a, b) ((a) > (b) ? (a) : (b))
#define sqr(x) ((x)*(x))
using namespace std;
const int N = 300;

int n, d;
double a[N+5][N+5];

void Gauss() {
  for (int line = 1; line <= n; line++) {
    int Max_line = line;
    for (int i = line+1; i <= n; i++)
      if (fabs(a[i][line]) > fabs(a[Max_line][line]))
    Max_line = i;
    if (Max_line != line) swap(a[line], a[Max_line]);
    if (a[line][line] == 0) {
      printf("No Solution
");
      return;
    }
    for (int i = line+1; i <= n; i++) {
      double tmp = a[i][line]/a[line][line];
      for (int j = line; j <= n+1; j++)
    a[i][j] -= a[line][j]*tmp;
    }
  }
  for (int i = n; i >= 1; i--) {
    for (int j = i+1; j <= n; j++)
      a[i][n+1] -= a[j][n+1]*a[i][j];
    a[i][n+1] /= a[i][i];
  }
  printf("%.4lf
", a[1][n+1]);
}
void work() {
  scanf("%d%d", &n, &d);
  memset(a, 0, sizeof(a));
  for (int i = 0; i < n ;i++) {
    if (i == d) {
      a[i+1][i+1] = 1;
      a[i+1][n+1] = 0;
    }else {
      a[i+1][i+1] = 1;
      a[i+1][(i-1+n)%n+1] = -0.5;
      a[i+1][(i+1)%n+1] = -0.5;
      a[i+1][n+1] = 1;
    }
  }
  Gauss();
}
int main() {
  int t; scanf("%d", &t);
  while (t--) work();
  return 0;
}