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# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Linq;
using System.Collections.Generic;
using System.Diagnostics;
using Debug = System.Diagnostics.Trace;
using SB = System.Text.StringBuilder;
using static System.Math;
using Number = System.Int32;

namespace Program {
	public class Solver {
		Random rnd = new Random();
		public void Solve() {
			var n = ri;
			var G = Enumerate(n, x => new List<KeyValuePair<int, long>>());
			for (int i = 0; i < n - 1; i++) {
				var u = ri - 1;
				var v = ri - 1;
				var w = ri;
				G[u].Add(new KeyValuePair<Number, long>(v, w));
				G[v].Add(new KeyValuePair<Number, long>(u, w));
			}
			var sz = new int[n];
			{
				Func<int, int, int> dfs = null;
				dfs = (prev, cur) => {
					var ret = 1;
					foreach (var e in G[cur])
						if (e.Key != prev) ret += dfs(cur, e.Key);
					return sz[cur] = ret;
				};
				dfs(-1, 0);
			}
			var c = new List<int>();
			for (int i = 0; i < n; i++) {
				var max = -1;
				foreach (var e in G[i]) {
					var s = sz[e.Key];
					if (s > sz[i]) s = n - sz[i];
					max = Max(max, s);
				}
				if (max * 2 <= n) c.Add(i);
			}
			Debug.WriteLine(c.AsJoinedString());
			var dist = new long[n];
			{
				Action<int, int, long> efs = null;
				efs = (prev, cur, d) => {
					dist[cur] = d;
					foreach (var e in G[cur])
						if (e.Key != prev) efs(cur, e.Key, d + e.Value);
				};
				if (c.Count == 1) foreach (var e in G[c[0]]) efs(c[0], e.Key, 0);
				else {
					efs(c[0], c[1], 0);
					efs(c[1], c[0], 0);
				}
			}
			Debug.WriteLine(dist.AsJoinedString());
			var ans = 2 * dist.Sum();
			if (c.Count == 2) ans += (n - 1) * G[c[0]].Where(x => x.Key == c[1]).First().Value;
			else {
				var mi = INF;
				var ma = 0L;
				foreach (var e in G[c[0]]) {
					var s = sz[e.Key];
					if (s > sz[c[0]]) s = n - sz[c[0]];
					ans += 2 * s * e.Value;
					Debug.WriteLine($"{e.Key} {2 * s * e.Value}");
					mi = Min(mi, e.Value);
				}
				ans += ma - mi;
			}
			Console.WriteLine(ans);

		}

		const long INF = 1L << 60;
		static int[] dx = { -1, 0, 1, 0 };
		static int[] dy = { 0, 1, 0, -1 };
		int ri { get { return sc.Integer(); } }
		long rl { get { return sc.Long(); } }
		double rd { get { return sc.Double(); } }
		string rs { get { return sc.Scan(); } }
		public IO.StreamScanner sc = new IO.StreamScanner(Console.OpenStandardInput());

		static T[] Enumerate<T>(int n, Func<int, T> f) {
			var a = new T[n];
			for (int i = 0; i < a.Length; ++i) a[i] = f(i);
			return a;
		}
		static public void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }
	}
}

#region main
static class Ex {
	static public string AsString(this IEnumerable<char> ie) { return new string(ie.ToArray()); }
	static public string AsJoinedString<T>(this IEnumerable<T> ie, string st = " ") {
		return string.Join(st, ie);
	}
	static public void Main() {
		Console.SetOut(new Program.IO.Printer(Console.OpenStandardOutput()) { AutoFlush = true });
		var solver = new Program.Solver();
		//* 
		var t = new System.Threading.Thread(solver.Solve, 50000000);
		t.Start();
		t.Join();
		//*/
		// solver.Solve();
		Console.Out.Flush();
	}
}
#endregion
#region Ex
namespace Program.IO {
	using System.IO;
	using System.Text;
	using System.Globalization;

	public class Printer : StreamWriter {
		public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
		public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { }
	}

	public class StreamScanner {
		public StreamScanner(Stream stream) { str = stream; }

		public readonly Stream str;
		private readonly byte[] buf = new byte[1024];
		private int len, ptr;
		public bool isEof = false;
		public bool IsEndOfStream { get { return isEof; } }

		private byte read() {
			if (isEof) return 0;
			if (ptr >= len) {
				ptr = 0;
				if ((len = str.Read(buf, 0, 1024)) <= 0) {
					isEof = true;
					return 0;
				}
			}
			return buf[ptr++];
		}

		public char Char() {
			byte b = 0;
			do b = read(); while ((b < 33 || 126 < b) && !isEof);
			return (char)b;
		}
		public string Scan() {
			var sb = new StringBuilder();
			for (var b = Char(); b >= 33 && b <= 126; b = (char)read()) sb.Append(b);
			return sb.ToString();
		}
		public string ScanLine() {
			var sb = new StringBuilder();
			for (var b = Char(); b != '\n' && b != 0; b = (char)read()) if (b != '\r') sb.Append(b);
			return sb.ToString();
		}
		public long Long() { return isEof ? long.MinValue : long.Parse(Scan()); }
		public int Integer() { return isEof ? int.MinValue : int.Parse(Scan()); }
		public double Double() { return isEof ? double.NaN : double.Parse(Scan(), CultureInfo.InvariantCulture); }
	}
}

#endregion