Tree and Permutation

Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 65536/65536 K (Java/Others)Total Submission(s): 0 Accepted Submission(s): 0

Problem Description

There are N vertices connected by N?1 edges, each edge has its own length.
The set { 1,2,3,…,N } contains a total of N! unique permutations, let’s say the i-th permutation is Pi and Pi,j is its j-th number.
For the i-th permutation, it can be a traverse sequence of the tree with N vertices, which means we can go from the Pi,1-th vertex to the Pi,2-th vertex by the shortest path, then go to the Pi,3-th vertex ( also by the shortest path ) , and so on. Finally we’ll reach the Pi,N-th vertex, let’s define the total distance of this route as D(Pi) , so please calculate the sum of D(Pi) for all N! permutations.

Input

There are 10 test cases at most.
The first line of each test case contains one integer N ( 1≤N≤105 ) .
For the next N?1 lines, each line contains three integer X, Y and L, which means there is an edge between X-th vertex and Y-th of length L ( 1≤X,Y≤N,1≤L≤109 ) .

Output

For each test case, print the answer module 109+7 in one line.

Sample Input

1 2 1

2 3 1

1 2 1

1 3 2

Sample Output

题意：

首先给出一个含有n个节点的树，边权为距离。

对于1-n的某一种排列p1,p2,p3……pn，贡献为dis(p1,p2)+dis(p2,p3)+dis(p3,p4)+……+dis(pn-1,pn)

求所有排列的贡献和

题解：

一棵树n个结点n-1条边，所以任意两点之间都有最短距离。由于所有遍历方式可以看做n个结点的排列，而每次有直接关系的只有两点之间的距离，因此只需求出两点之间的距离在所有排列中一共出现多少次即可。

比如n=2时，两个点1,2，共有两种排列方式，即1,2,和2,1

n=3时，三个点1,2,3,其中1,2共有4种方法得到，即1,2,3,和2,1,3,和3,1,2,和3,2,1

n=4时，三个点1,2,3,4,其中1,2共有12种方法得到，即1,2,3,4,和1,2,4,3,和2,1,3,4,和2,1,4,3,和3,1,2,4,和3,2,1,4和4,1,2,3,和4,2,1,3,和3,4,1,2,和4,3,1,2,和3,4,2,1,和4,3,2,1,

n=jie[n-1]*（n-1）。

先用树状dp求出所有两两结点之间的距离之和，再乘以每条边在所有结点排列的n！方式中一共用到的次数。

代码：

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const int maxn = 1e5+7;
const ll mod = 1e9+7;
ll sum[maxn], n;
ll dp[maxn];
ll jie[maxn];
ll jie2[maxn];
struct Edge
{
    ll v, w;
};
vector<Edge> tree[maxn];
void init(){
    jie[2] = 2;
    for(ll i = 3;i<maxn;i++){
        jie[i] = jie[i-1]*(i-1)%mod;
    }
}
void dfs(ll cur, ll father)
{
    sum[cur] = 1;
    for(ll i = 0; i < tree[cur].size(); i++)
    {
        ll son = tree[cur][i].v;
        ll len = tree[cur][i].w;
        if(father == son)
            continue;
        dfs(son, cur);
        sum[cur] += sum[son]%mod;
        sum[cur]%=mod;
        dp[cur] += (dp[son]%mod + (n-sum[son])%mod*sum[son]%mod * len%mod)%mod;
        dp[cur]%=mod;
    }
}
int main()
{
    init();
    ll u, v, w;
    while(scanf("%lld", &n)!=EOF)
    {
        if(n<=2){
            if(n==1){
                printf("0\n");
                continue;
            }
            scanf("%lld%lld%lld", &u, &v, &w);
            printf("%lld\n", 2*w%mod);
            continue;
        }
        for(ll i = 0; i < n; i++)
            tree[i].clear();
        memset(sum, 0, sizeof(sum));
        memset(dp, 0, sizeof(dp));
        for(ll i = 0; i < n-1; i++)
        {
            scanf("%lld%lld%lld", &u, &v, &w);
            u--,v--;
            Edge t1, t2;
            t1.v = v;
            t1.w = w;
            t2.v = u;
            t2.w = w;
            tree[u].push_back(t1);
            tree[v].push_back(t2);
        }
        dfs(0, -1);
        printf("%lld\n", (dp[0]%mod)*jie[n]%mod);
    }
    return 0;
}