CF1000G.Two-Paths

You are given a weighted tree (undirected connected graph with no cycles, loops or multiple edges) with $n$ vertices. The edge $\{u_j, v_j\}$ has weight $w_j$ . Also each vertex $i$ has its own value $a_i$ assigned to it.

Let's call a path starting in vertex $u$ and ending in vertex $v$ , where each edge can appear no more than twice (regardless of direction), a 2-path. Vertices can appear in the 2-path multiple times (even start and end vertices).

For some 2-path $p$ profit $\text{Pr}(p) = \sum\limits_{v \in \text{distinct vertices in } p}{a_v} - \sum\limits_{e \in \text{distinct edges in } p}{k_e \cdot w_e}$ , where $k_e$ is the number of times edge $e$ appears in $p$ . That is, vertices are counted once, but edges are counted the number of times they appear in $p$ .

You are about to answer $m$ queries. Each query is a pair of vertices $(qu, qv)$ . For each query find 2-path $p$ from $qu$ to $qv$ with maximal profit $\text{Pr}(p)$ .

The first line contains two integers $n$ and $q$ ( $2 \le n \le 3 \cdot 10^5$ , $1 \le q \le 4 \cdot 10^5$ ) — the number of vertices in the tree and the number of queries.

The second line contains $n$ space-separated integers $a_1, a_2, \dots, a_n$ $(1 \le a_i \le 10^9)$ — the values of the vertices.

Next $n - 1$ lines contain descriptions of edges: each line contains three space separated integers $u_i$ , $v_i$ and $w_i$ ( $1 \le u_i, v_i \le n$ , $u_i \neq v_i$ , $1 \le w_i \le 10^9$ ) — there is edge $\{u_i, v_i\}$ with weight $w_i$ in the tree.

Next $q$ lines contain queries (one per line). Each query contains two integers $qu_i$ and $qv_i$ $(1 \le qu_i, qv_i \le n)$ — endpoints of the 2-path you need to find.

For each query print one integer per line — maximal profit $\text{Pr}(p)$ of the some 2-path $p$ with the corresponding endpoints.

Explanation of queries:

1. $(1, 1)$ — one of the optimal 2-paths is the following: $1 \rightarrow 2 \rightarrow 4 \rightarrow 5 \rightarrow 4 \rightarrow 2 \rightarrow 3 \rightarrow 2 \rightarrow 1$ . $\text{Pr}(p) = (a_1 + a_2 + a_3 + a_4 + a_5) - (2 \cdot w(1,2) + 2 \cdot w(2,3) + 2 \cdot w(2,4) + 2 \cdot w(4,5)) = 21 - 2 \cdot 12 = 9$ .
2. $(4, 4)$ : $4 \rightarrow 2 \rightarrow 1 \rightarrow 2 \rightarrow 3 \rightarrow 2 \rightarrow 4$ . $\text{Pr}(p) = (a_1 + a_2 + a_3 + a_4) - 2 \cdot (w(1,2) + w(2,3) + w(2,4)) = 19 - 2 \cdot 10 = 9$ .
3. $(5, 6)$ : $5 \rightarrow 4 \rightarrow 2 \rightarrow 3 \rightarrow 2 \rightarrow 1 \rightarrow 2 \rightarrow 4 \rightarrow 6$ .
4. $(6, 4)$ : $6 \rightarrow 4 \rightarrow 2 \rightarrow 1 \rightarrow 2 \rightarrow 3 \rightarrow 2 \rightarrow 4$ .
5. $(3, 4)$ : $3 \rightarrow 2 \rightarrow 1 \rightarrow 2 \rightarrow 4$ .
6. $(3, 7)$ : $3 \rightarrow 2 \rightarrow 1 \rightarrow 2 \rightarrow 4 \rightarrow 5 \rightarrow 4 \rightarrow 2 \rightarrow 3 \rightarrow 7$ .