Problem:
Score : 1000 points

Problem Statement

You are given a tree with N vertices. The vertices are numbered 0 through N-1, and the edges are numbered 1 through N-1.
Edge i connects Vertex x_i and y_i, and has a value a_i.
You can perform the following operation any number of times:

Choose a simple path and a non-negative integer x, then for each edge e that belongs to the path, change a_e by executing a_e ← a_e ⊕ x (⊕ denotes XOR).

Your objective is to have a_e = 0 for all edges e.
Find the minimum number of operations required to achieve it.

Constraints

2 ≤ N ≤ 10^5

0 ≤ x_i,y_i ≤ N-1

0 ≤ a_i ≤ 15

The given graph is a tree.

All input values are integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1 a_1
x_2 y_2 a_2
:
x_{N-1} y_{N-1} a_{N-1}

Output

Find the minimum number of operations required to achieve the objective.

Sample Input 1

5
0 1 1
0 2 3
0 3 6
3 4 4

Sample Output 1

3

The objective can be achieved in three operations, as follows:

First, choose the path connecting Vertex 1, 2, and x = 1.

Then, choose the path connecting Vertex 2, 3, and x = 2.

Lastly, choose the path connecting Vertex 0, 4, and x = 4.

Sample Input 2

2
1 0 0

Sample Output 2

0

Solution in C++14 (GCC 5.4.1) (judge verdict: Runtime Error)

//#include<bits/stdc++.h>
#include<iostream>
#include <vector>
#include <algorithm>
#include <cmath>
#include <set>
#include <queue>
#include <deque>
#include <map>
#include <stack>
#include<bitset>
#include<list>
#include<cassert>
#include<numeric>

using namespace std;

const int N = 1e5 + 5;
const int M = 20;

int n, m;
int a[N];
int ans;
int cnt[M];
int dp[1 << M];
int dp2[1 << M];
vector<int> v;

int main()
{
	cin >> n;
	for (int i = 1; i < n; i++)
	{
		int u, v, x;
		cin >> v >> u >> x;
		a[u] = a[u] ^ x;
		a[v] = a[v] ^ x;
	}
	for (int i = 0; i < n; i++)
	{
		cnt[a[i]]++;
	}
	ans = cnt[0];
	for (int i = 1; i < n; i++)
	{
		if (cnt[a[i]] % 2)
		{
			v.push_back(a[i]);
		}
		ans += cnt[a[i]] / 2;
	}
	m = v.size();
	for (int i = 0; i < (1 << m); i++)
	{
		for (int j = 0; j < m; j++)
		{
			if ((i >> j) % 2 == 1)
			{
				dp2[i] ^= v[j];
			}
		}
	}
	for (int mask = 1; mask < (1 << m); mask++)
	{
		if (dp2[mask] != 0)
		{
			continue;
		}
		dp[mask] = 1;
		int mask2 = mask;
		while (mask2 > 0)
		{
			mask2 = (mask2 - 1) & mask;
			if (!dp2[mask2])
			{
				dp[mask] = max(dp[mask], dp[mask2] + dp[mask ^ mask2]);
			}
		}
	}
	cout << max(n - ans - dp[(1 << m) - 1], 0) << endl;
}