Select Edges

AtCoder

IDabc259_f

Time3000ms

Memory256MB

Difficulty—

You are given a tree with $N$ vertices. For each $i = 1, 2, \ldots, N-1$, the $i$\-th edge connects Vertex $u_i$ and Vertex $v_i$ and has a weight $w_i$. Consider choosing some of the $N-1$ edges (possibly none or all). Here, for each $i = 1, 2, \ldots, N$, one may choose at most $d_i$ edges incident to Vertex $i$. Find the maximum possible total weight of the chosen edges. ## Constraints * $2 \leq N \leq 3 \times 10^5$ * $1 \leq u_i, v_i \leq N$ * $-10^9 \leq w_i \leq 10^9$ * $d_i$ is a non-negative integer not exceeding the degree of Vertex $i$. * The given graph is a tree. * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $d_1$ $d_2$ $\ldots$ $d_N$ $u_1$ $v_1$ $w_1$ $u_2$ $v_2$ $w_2$ $\vdots$ $u_{N-1}$ $v_{N-1}$ $w_{N-1}$ [samples]

Samples

Input #1

7
1 2 1 0 2 1 1
1 2 8
2 3 9
2 4 10
2 5 -3
5 6 8
5 7 3

Output #1

28

If you choose the $1$\-st, $2$\-nd, $5$\-th, and $6$\-th edges, the total weight of those edges is $8 + 9 + 8 + 3 = 28$. This is the maximum possible.

Input #2

20
0 2 0 1 2 1 0 0 3 0 1 1 1 1 0 0 3 0 1 2
4 9 583
4 6 -431
5 9 325
17 6 131
17 2 -520
2 16 696
5 7 662
17 15 845
7 8 307
13 7 849
9 19 242
20 6 909
7 11 -775
17 18 557
14 20 95
18 10 646
4 3 -168
1 3 -917
11 12 30

Output #2

API Response (JSON)

{
  "problem": {
    "name": "Select Edges",
    "description": {
      "content": "You are given a tree with $N$ vertices. For each $i = 1, 2, \\ldots, N-1$, the $i$\\-th edge connects Vertex $u_i$ and Vertex $v_i$ and has a weight $w_i$. Consider choosing some of the $N-1$ edges (pos",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc259_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a tree with $N$ vertices. For each $i = 1, 2, \\ldots, N-1$, the $i$\\-th edge connects Vertex $u_i$ and Vertex $v_i$ and has a weight $w_i$.\nConsider choosing some of the $N-1$ edges (pos...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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