Expensive Expense

AtCoder

IDabc222_f

Time4000ms

Memory256MB

Difficulty—

The Kingdom of AtCoder is composed of $N$ towns and $N-1$ roads. The towns are labeled as Town $1$, Town $2$, $\dots$, Town $N$. Similarly, the roads are labeled as Road $1$, Road $2$, $\dots$, Road $N-1$. Road $i$ connects Towns $A_i$ and $B_i$ bidirectionally, and you have to pay the toll of $C_i$ to go through it. For every pair of different towns $(i, j)$, it is possible to go from Town $A_i$ to Town $B_j$ via the roads. You are given a sequence $D = (D_1, D_2, \dots, D_N)$, where $D_i$ is the cost to do sightseeing in Town $i$. Let us define the travel cost $E_{i,j}$ from Town $i$ to Town $j$ as the total toll incurred when going from Town $i$ to Town $j$, plus $D_{j}$. * More formally, $E_{i,j}$ is defined as $D_j + \displaystyle\sum_{l=0}^{k-1} c_l$, where the shortest path between $i$ and $j$ is $i = p_0, p_1, \dots, p_{k-1}, p_k = j$ and the toll for the road connecting Towns $p_{l}$ and $p_{l+1}$ is $c_l$. For every $i$, find the maximum possible travel cost when traveling from Town $i$ to another town. * More formally, for every $i$, find $\max_{1 \leq j \leq N, j \neq i} E_{i,j}$. ## Constraints * $2 \leq N \leq 2 \times 10^5$ * $1 \leq A_i \leq N$ $(1 \leq i \leq N-1)$ * $1 \leq B_i \leq N$ $(1 \leq i \leq N-1)$ * $1 \leq C_i \leq 10^9$ $(1 \leq i \leq N-1)$ * $1 \leq D_i \leq 10^9$ $(1 \leq i \leq N)$ * It is possible to travel from Town $i$ to Town $j$ via some number of roads, for a pair of integers $(i,j)$ such that $1 \leq i \lt j \leq N$. * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $A_1$ $B_1$ $C_1$ $A_2$ $B_2$ $C_2$ $\vdots$ $A_{N-1}$ $B_{N-1}$ $C_{N-1}$ $D_1$ $D_2$ $\dots$ $D_N$ [samples]

Samples

Input #1

Output #1

8
6
6

The value of $E_{i,j}$ for every ordered pair of towns $(i,j)$ is as follows.

*   $E_{1,2} = 2 + 2 = 4$
*   $E_{1,3} = 5 + 3 = 8$
*   $E_{2,1} = 2 + 1 = 3$
*   $E_{2,3} = 3 + 3 = 6$
*   $E_{3,1} = 5 + 1 = 6$
*   $E_{3,2} = 3 + 2 = 5$

Input #2

6
1 2 3
1 3 1
1 4 4
1 5 1
1 6 5
9 2 6 5 3 100

Output #2

Input #3

6
1 2 1000000000
2 3 1000000000
3 4 1000000000
4 5 1000000000
5 6 1000000000
1 2 3 4 5 6

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Expensive Expense",
    "description": {
      "content": "The Kingdom of AtCoder is composed of $N$ towns and $N-1$ roads.   The towns are labeled as Town $1$, Town $2$, $\\dots$, Town $N$. Similarly, the roads are labeled as Road $1$, Road $2$, $\\dots$, Road",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 4000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc222_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "The Kingdom of AtCoder is composed of $N$ towns and $N-1$ roads.  \nThe towns are labeled as Town $1$, Town $2$, $\\dots$, Town $N$. Similarly, the roads are labeled as Road $1$, Road $2$, $\\dots$, Road...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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