Problem:
Score : 1100 points

Problem Statement

There is a tree with N vertices, numbered 1 through N.
The i-th edge in this tree connects Vertices A_i and B_i and has a length of C_i.

Joisino created a complete graph with N vertices.
The length of the edge connecting Vertices u and v in this graph, is equal to the shortest distance between Vertices u and v in the tree above.

Joisino would like to know the length of the longest Hamiltonian path (see Notes) in this complete graph.
Find the length of that path.

Notes

A Hamiltonian path in a graph is a path in the graph that visits each vertex exactly once.

Constraints

2 \leq N \leq 10^5

1 \leq A_i < B_i \leq N

The given graph is a tree.

1 \leq C_i \leq 10^8

All input values are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 B_1 C_1
A_2 B_2 C_2
:
A_{N-1} B_{N-1} C_{N-1}

Output

Print the length of the longest Hamiltonian path in the complete graph created by Joisino.

Sample Input 1

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

Sample Output 1

38

The length of the Hamiltonian path 5 → 3 → 1 → 4 → 2 is 5+8+15+10=38. Since there is no Hamiltonian path with length 39 or greater in the graph, the answer is 38.

Sample Input 2

8
2 8 8
1 5 1
4 8 2
2 5 4
3 8 6
6 8 9
2 7 12

Sample Output 2

132

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <algorithm>
#include <iostream>
#include <vector>
using namespace std;

const int MAXN = 1e5 + 10, MAXV = 1e9;
vector<pair<int, int> > Mat[MAXN];
long long ans, val[MAXN];
int a[MAXN], b[MAXN], n;
int dfs(int v, int p, int w) {
	int cnt = 1;
	for (int i = 0; i < Mat[v].size(); i++) {
		int u = Mat[v][i].first, o = Mat[v][i].second;
		if (u != p)
			cnt += dfs(u, v, o);
	}
	ans += min(cnt, n - cnt) * 2 * w;
	if (cnt == n - cnt)
		val[1] += w;
	else if (cnt < n - cnt)
		val[v] += w;
	else
		val[1] += w, val[v] -= w;
	return cnt;
}
void dfso(int v, int p, long long tmp) {
	val[v] += tmp;
	for (int i = 0; i < Mat[v].size(); i++) {
		int u = Mat[v][i].first, o = Mat[v][i].second;
		if (u != p)
			dfso(u, v, val[v]);
	}
}

int main() {
	cin >> n;
	for (int i = 0; i < n - 1; i++) {
		int a, b, w;
		cin >> a >> b >> w;
		Mat[a].push_back({ b, w });
		Mat[b].push_back({ a, w });
	}
	dfs(1, 1, 0);
	dfso(1, 1, 0);
	long long mn[2] = { MAXV, MAXV };
	for (int i = 1; i <= n; i++) {
		mn[1] = min(mn[1], val[i]);
		if (mn[1] < mn[0])
			swap(mn[1], mn[0]);
	}
	cout << ans - max(mn[0], mn[1]);
}