HDU - 5036 Explosion

Posted 2020-06-13

Problem Description

Everyone knows Matt enjoys playing games very much. Now, he is playing such a game. There are N rooms, each with one door. There are some keys(could be none) in each room corresponding to some doors among these N doors. Every key can open only one door. Matt has some bombs, each of which can destroy a door. He will uniformly choose a door that can not be opened with the keys in his hand to destroy when there are no doors that can be opened with keys in his hand. Now, he wants to ask you, what is the expected number of bombs he will use to open or destroy all the doors. Rooms are numbered from 1 to N.

 

Input

The first line of the input contains an integer T, denoting the number of testcases. Then T test cases follow.

In the first line of each test case, there is an integer N (N<=1000) indicating the number of rooms. 

The following N lines corresponde to the rooms from 1 to N. Each line begins with an integer k (0<=k<=N) indicating the number of keys behind the door. Then k integers follow corresponding to the rooms these keys can open.

 

Output

For each test case, output one line "Case #x: y", where x is the case number (starting from 1), y is the answer which should be rounded to 5 decimal places.

 

Sample Input

2
3
1 2
1 3
1 1
3
0
0
0

 

Sample Output

Case #1: 1.00000
Case #2: 3.00000

 

Source

php?

field=problem&key=2014+ACM%2FICPC+Asia+Regional+Beijing+Online&source=1&searchmode=source" style="color:rgb(26,92,200); text-decoration:none">2014 ACM/ICPC Asia Regional Beijing Online

题意：n个房间，每一个房间都有若干个钥匙打开其它的门，假设手上没有钥匙能够选择等概率随机选择一个门炸开。求用炸弹数的期望。

思路：每一个房间期望都是可加的。

单独考虑一个房间，假设有k个房间被炸开都会导致这个房间被打开。

那么炸一次这个房间被打开的概率即为kn。也就意味着为了把这个房间打开的期望炸的次数为nk。所有加起来后除以n就可以。bitset优化。

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <bitset>
using namespace std;
const int maxn = 1005;

bitset<maxn> a[maxn];

int main() {
	int t, cas = 1;
	int n;
	scanf("%d", &t);
	while (t--) {
		scanf("%d", &n);
		for (int i = 0; i < n; i++) {
			a[i].reset();
			a[i][i] = 1;
		}

		int c, x;
		for (int i = 0; i < n; i++) {
			scanf("%d", &c);
			while (c--) {
				scanf("%d", &x);
				a[i][--x] = 1;
			}
		}

		for (int i = 0; i < n; i++)
			for (int j = 0; j < n; j++) 
				if (a[j][i])
					a[j] |= a[i];

		double ans = 0;
		for (int i = 0; i < n; i++) {
			c = 0;
			for (int j = 0; j < n; j++)
				if (a[j][i]) 
					c++;
			ans += 1.0 / c;
		}

		printf("Case #%d: %.5lf\n",cas++,ans);
	}
	return 0;
}