P3714 [BJOI2017] A Hard Problem on Trees

You are given an unrooted tree with $n$ nodes. Each edge of the tree has a color. There are $m$ colors, numbered from $1$ to $m$, and the weight of the $i$-th color is $c_i$. For a simple path on the tree, the edges along the path form a color sequence in order, which can be split into several segments of the same color. Define the path value as the sum of the color weights of each same-color segment in the sequence. Please compute the maximum path value among all simple paths whose number of edges is between $l$ and $r$.

The first line contains four integers $n, m, l, r$. The second line contains $m$ integers $c_1, c_2, \ldots, c_m$, separated by spaces, representing the weight of each color in order. The next $n-1$ lines each contain three integers $u, v, c$, indicating that there is an edge between node $u$ and node $v$ with color $c$.

Output one line with a single integer, the answer.

Sample explanation 1: All color weights are negative, and the optimal path is $(1, 2)$ or $(1, 3)$. Sample explanation 2: The optimal path is $(3, 1, 2, 5, 6)$, and its color sequence is $(2, 1, 1, 2)$. Constraints | Testpoint ID | $n$ | $m$ | Special constraints | |-|-|-|-| | $1$ | $=10^3$ | $\le n$ | None | | $2$ | $=10^4$ | $=2$ | None | | $3$ | $=10^5$ | $\le n$ | The degree of every node is at most $2$ | | $4$ | $=2\times10^5$ | $\le n$ | The degree of every node is at most $2$ | | $5$ | $=10^5$ | $=10$ | $l=1$, $r=n-1$ | | $6$ | $=2\times10^5$ | $\le n$ | $l=1$, $r=n-1$ | | $7$ | $=10^5$ | $=50$ | None | | $8$ | $=10^5$ | $\le n$ | None | | $9$ | $=2\times 10^5$ | $=100$ | None | | $10$ | $=2\times 10^5$ | $\le n$ | None | For $100\%$ of the testdata, $1 \leq n, m \leq 2 \times 10^5$, $1 \leq l \leq r \leq n$, and $\lvert c_i \rvert \leq 10^4$. It is guaranteed that there exists at least one path whose number of edges is between $l$ and $r$. Translated by ChatGPT 5