Minimize Sum of Distances

AtCoder

IDabc348_e

Time2000ms

Memory256MB

Difficulty—

You are given a tree with $N$ vertices. The vertices are numbered $1$ to $N$, and the $i$\-th edge connects vertices $A_i$ and $B_i$. You are also given a sequence of positive integers $C = (C_1, C_2, \ldots ,C_N)$ of length $N$. Let $d(a, b)$ be the number of edges between vertices $a$ and $b$, and for $x = 1, 2, \ldots, N$, let $\displaystyle f(x) = \sum_{i=1}^{N} (C_i \times d(x, i))$. Find $\displaystyle \min_{1 \leq v \leq N} f(v)$. ## Constraints * $1 \leq N \leq 10^5$ * $1 \leq A_i, B_i \leq N$ * The given graph is a tree. * $1 \leq C_i \leq 10^9$ ## Input The input is given from Standard Input in the following format: $N$ $A_1$ $B_1$ $A_2$ $B_2$ $\vdots$ $A_{N - 1}$ $B_{N - 1}$ $C_1$ $C_2$ $\cdots$ $C_N$ [samples]

Samples

Input #1

Output #1

5

For example, consider calculating $f(1)$. We have $d(1, 1) = 0, d(1, 2) = 1, d(1, 3) = 1, d(1, 4) = 2$.  
Thus, $f(1) = 0 \times 1 + 1 \times 1 + 1 \times 1 + 2 \times 2 = 6$.
Similarly, $f(2) = 5, f(3) = 9, f(4) = 6$. Since $f(2)$ is the minimum, print `5`.

Input #2

2
2 1
1 1000000000

Output #2

1

$f(2) = 1$, which is the minimum.

Input #3

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

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Minimize Sum of Distances",
    "description": {
      "content": "You are given a tree with $N$ vertices. The vertices are numbered $1$ to $N$, and the $i$\\-th edge connects vertices $A_i$ and $B_i$. You are also given a sequence of positive integers $C = (C_1, C_2,",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc348_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a tree with $N$ vertices. The vertices are numbered $1$ to $N$, and the $i$\\-th edge connects vertices $A_i$ and $B_i$.\nYou are also given a sequence of positive integers $C = (C_1, C_2,...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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