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F - Select Edges

Score : $500$ points

Problem Statement

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:

$N$

$d_1$ $d_2$ $\ldots$ $d_N$

$u_1$ $v_1$ $w_1$

$u_2$ $v_2$ $w_2$

$\vdots$

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

Output

Print the answer.

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

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.

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

2184