Good Bye 2014 D. New Year Santa Network 图论+期望

D. New Year Santa Network

 

New Year is coming in Tree World! In this world, as the name implies, there are n cities connected by n - 1 roads, and for any two distinct cities there always exists a path between them. The cities are numbered by integers from 1 to n, and the roads are numbered by integers from 1 to n - 1. Let's define d(u, v) as total length of roads on the path between city u and city v.

As an annual event, people in Tree World repairs exactly one road per year. As a result, the length of one road decreases. It is already known that in the i-th year, the length of the ri-th road is going to become wi, which is shorter than its length before. Assume that the current year is year 1.

Three Santas are planning to give presents annually to all the children in Tree World. In order to do that, they need some preparation, so they are going to choose three distinct cities c1, c2, c3 and make exactly one warehouse in each city. The k-th (1 ≤ k ≤ 3) Santa will take charge of the warehouse in city ck.

It is really boring for the three Santas to keep a warehouse alone. So, they decided to build an only-for-Santa network! The cost needed to build this network equals to d(c1, c2) + d(c2, c3) + d(c3, c1) dollars. Santas are too busy to find the best place, so they decided to choose c1, c2, c3 randomly uniformly over all triples of distinct numbers from 1 to n. Santas would like to know the expected value of the cost needed to build the network.

However, as mentioned, each year, the length of exactly one road decreases. So, the Santas want to calculate the expected after each length change. Help them to calculate the value.

Input

The first line contains an integer n (3 ≤ n ≤ 105) — the number of cities in Tree World.

Next n - 1 lines describe the roads. The i-th line of them (1 ≤ i ≤ n - 1) contains three space-separated integers ai, bi, li (1 ≤ ai, bi ≤ n,ai ≠ bi, 1 ≤ li ≤ 103), denoting that the i-th road connects cities ai and bi, and the length of i-th road is li.

The next line contains an integer q (1 ≤ q ≤ 105) — the number of road length changes.

Next q lines describe the length changes. The j-th line of them (1 ≤ j ≤ q) contains two space-separated integers rj, wj (1 ≤ rj ≤ n - 1,1 ≤ wj ≤ 103). It means that in the j-th repair, the length of the rj-th road becomes wj. It is guaranteed that wj is smaller than the current length of the rj-th road. The same road can be repaired several times.

Output

Output q numbers. For each given change, print a line containing the expected cost needed to build the network in Tree World. The answer will be considered correct if its absolute and relative error doesn't exceed 10 - 6.

Sample test(s)

input

3
2 3 5
1 3 3
5
1 4
2 2
1 2
2 1
1 1

output

14.0000000000
12.0000000000
8.0000000000
6.0000000000
4.0000000000

input

6
1 5 3
5 3 2
6 1 7
1 4 4
5 2 3
5
1 2
2 1
3 5
4 1
5 2

output

19.6000000000
18.6000000000
16.6000000000
13.6000000000
12.6000000000

Note

Consider the first sample. There are 6 triples: (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1). Because n = 3, the cost needed to build the network is always d(1, 2) + d(2, 3) + d(3, 1) for all the triples. So, the expected cost equals to d(1, 2) + d(2, 3) + d(3, 1).

 题意：

给你一颗树，边有权值，现在让你计算任意三个不同的点的距离和的期望

题解：

对于任取三个点，我们可以知道有sum=n*(n-1)*(n-2)/6中取法，

我们对于到达x的边序号id，x及其子树节点数为size[x],那么  对于经过x这个点的取法就有  tmp=size[x]*(n-size[x])*(n-2)

对于使用id这条边其概率就是  pid= tmp/sum

期望就是 w[i]*pid

#include <bits/stdc++.h>
using namespace std;
#define rep(i,a,n) for (int i=a;i<n;i++)
#define per(i,a,n) for (int i=n-1;i>=a;i--)
#define pb push_back
#define mp make_pair
#define all(x) (x).begin(),(x).end()
#define SZ(x) ((int)(x).size())
#define fi first
#define se second
typedef vector<int> VI;
typedef long long ll;
typedef pair<int,int> PII;
const ll mod=;
ll powmod(ll a,ll b) {ll res=;a%=mod;for(;b;b>>=){if(b&)res=res*a%mod;a=a*a%mod;}return res;}
// head
 
const int N=;
int sz[N],w[N],n,u,v,m;
vector<PII> e[N];
double prob[N],ret;
void dfs(int u,int f,int id) {
    sz[u]=;
    rep(j,,SZ(e[u])) {
        int v=e[u][j].fi;
        if (v==f) continue;
        dfs(v,u,e[u][j].se);
        sz[u]+=sz[v];
    }
    prob[id]=6.0*sz[u]*(n-sz[u])/n/(n-);
}
int main() {
    scanf("%d",&n);
    rep(i,,n) {
        scanf("%d%d%d",&u,&v,w+i);
        e[u].pb(mp(v,i));
        e[v].pb(mp(u,i));
    }
    dfs(,,);
    //for(int i=1;i<=n;i++) cout<<prob[i]<<" ";cout<<endl;
    rep(i,,n) ret+=w[i]*prob[i];
    scanf("%d",&m);
    rep(i,,m) {
        scanf("%d%d",&u,&v);
        v=w[u]-v;
        ret-=v*prob[u];
        printf("%.10f\n",ret);
        w[u]-=v;
    }
}

代码