Description

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 $r$.

Input Format

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 Format

Output one line with a single integer, the answer.

5 3 1 4
-1 -5 -2
1 2 1
1 3 1
2 4 2
2 5 3

-1

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

11

Hint

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