Score : $400$ points

问题描述

我们有一个包含 $N$ 个顶点的树，顶点编号为 $1, 2, \dots, N$。
第 $i$ 条边（$1 \leq i \leq N - 1$）连接顶点 $u_i$ 和顶点 $v_i$，并具有权重 $w_i$。

对于不同的顶点 $u$ 和 $v$，令 $f(u, v)$ 表示从顶点 $u$ 到顶点 $v$ 的最短路径中包含的最大权重边。

求 $\displaystyle \sum_{i = 1}^{N - 1} \sum_{j = i + 1}^N f(i, j)$。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

We have a tree with $N$ vertices numbered $1, 2, \dots, N$.
The $i$-th edge $(1 \leq i \leq N - 1)$ connects Vertex $u_i$ and Vertex $v_i$ and has a weight $w_i$.

For different vertices $u$ and $v$, let $f(u, v)$ be the greatest weight of an edge contained in the shortest path from Vertex $u$ to Vertex $v$.

Find $\displaystyle \sum_{i = 1}^{N - 1} \sum_{j = i + 1}^N f(i, j)$.

Constraints

• $2 \leq N \leq 10^5$
• $1 \leq u_i, v_i \leq N$
• $1 \leq w_i \leq 10^7$
• The given graph is a tree.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$u_1$ $v_1$ $w_1$

$\vdots$

$u_{N - 1}$ $v_{N - 1}$ $w_{N - 1}$

Output

Print the answer.

Sample Input 1

3
1 2 10
2 3 20

Sample Output 1

50

We have $f(1, 2) = 10$, $f(2, 3) = 20$, and $f(1, 3) = 20$, so we should print their sum, or $50$.

Sample Input 2

5
1 2 1
2 3 2
4 2 5
3 5 14

Sample Output 2

76

update @ 2024/3/10 09:31:34