The Child and Zoo
题意翻译
我们的小朋友们十分喜欢在动物园里散步。动物园由 $n$ 个从 $1$ 至 $n$ 编号的区域组成。第 $i$ 个区域里有 $a_i$ 只动物。动物园中有 $m$ 条双向道路连接两个区域。当然动物园是联通的，所以你可以从一个区域走到任何其他区域。
我们的小朋友十分聪明。如果一个小朋友想从 $p$ 区域走到 $q$ 区域，首先，他会把所有从 $p$ 至 $q$ 的简单路径经过区域的最小值记录下来，然后从中找出最大值，记做 $f(p,q)$。最后，小朋友会挑选一条经过区域最小值为 $f(p,q)$ 的路径。
当小朋友参观完动物园后，他想出了一个问题：所有 $f(p,q)$ $(p \neq q)$ 的平均值是多少？
题目描述
Of course our child likes walking in a zoo. The zoo has $ n $ areas, that are numbered from $ 1 $ to $ n $ . The $ i $ -th area contains $ a_{i} $ animals in it. Also there are $ m $ roads in the zoo, and each road connects two distinct areas. Naturally the zoo is connected, so you can reach any area of the zoo from any other area using the roads.
Our child is very smart. Imagine the child want to go from area $ p $ to area $ q $ . Firstly he considers all the simple routes from $ p $ to $ q $ . For each route the child writes down the number, that is equal to the minimum number of animals among the route areas. Let's denote the largest of the written numbers as $ f(p,q) $ . Finally, the child chooses one of the routes for which he writes down the value $ f(p,q) $ .
After the child has visited the zoo, he thinks about the question: what is the average value of $ f(p,q) $ for all pairs $ p,q $ $ (p≠q) $ ? Can you answer his question?
输入输出格式
输入格式
The first line contains two integers $ n $ and $ m $ ( $ 2<=n<=10^{5} $ ; $ 0<=m<=10^{5} $ ). The second line contains $ n $ integers: $ a_{1},a_{2},...,a_{n} $ ( $ 0<=a_{i}<=10^{5} $ ). Then follow $ m $ lines, each line contains two integers $ x_{i} $ and $ y_{i} $ ( $ 1<=x_{i},y_{i}<=n $ ; $ x_{i}≠y_{i} $ ), denoting the road between areas $ x_{i} $ and $ y_{i} $ .
All roads are bidirectional, each pair of areas is connected by at most one road.
输出格式
Output a real number — the value of .
The answer will be considered correct if its relative or absolute error doesn't exceed $ 10^{-4} $ .
输入输出样例
输入样例 #1
4 3
10 20 30 40
1 3
2 3
4 3
输出样例 #1
16.666667
输入样例 #2
3 3
10 20 30
1 2
2 3
3 1
输出样例 #2
13.333333
输入样例 #3
7 8
40 20 10 30 20 50 40
1 2
2 3
3 4
4 5
5 6
6 7
1 4
5 7
输出样例 #3
18.571429
说明
Consider the first sample. There are $ 12 $ possible situations:
- $ p=1,q=3,f(p,q)=10 $ .
- $ p=2,q=3,f(p,q)=20 $ .
- $ p=4,q=3,f(p,q)=30 $ .
- $ p=1,q=2,f(p,q)=10 $ .
- $ p=2,q=4,f(p,q)=20 $ .
- $ p=4,q=1,f(p,q)=10 $ .
Another $ 6 $ cases are symmetrical to the above. The average is .
Consider the second sample. There are $ 6 $ possible situations:
- $ p=1,q=2,f(p,q)=10 $ .
- $ p=2,q=3,f(p,q)=20 $ .
- $ p=1,q=3,f(p,q)=10 $ .
Another $ 3 $ cases are symmetrical to the above. The average is .