#include <bits/stdc++.h>
using namespace std;
vector<pair<int, long long> > nei[300005];
int fa[20][300005], dep[300005];
long long w[300005], ma[300005];
void dfs(int v, int pa, int depth) {
  fa[0][v] = pa;
  ma[v] = w[v];
  dep[v] = depth;
  for (int i = 0; i < nei[v].size(); i++) {
    int u = nei[v][i].first;
    long long co = nei[v][i].second;
    if (u == pa) continue;
    dfs(u, v, depth + 1);
    ma[v] = max(ma[v] + ma[u] - 2 * co, ma[v]);
  }
}
long long dp[300005], dp2[300005];
void dfs2(int v, int pa) {
  long long sum = 0;
  for (int i = 0; i < nei[v].size(); i++) {
    int u = nei[v][i].first;
    long long co = nei[v][i].second;
    if (u == pa) continue;
    if (ma[u] - 2 * co > 0) sum += ma[u] - 2 * co;
  }
  for (int i = 0; i < nei[v].size(); i++) {
    int u = nei[v][i].first;
    long long co = nei[v][i].second;
    if (u == pa) continue;
    if (ma[u] - 2 * co > 0) {
      dp2[u] = w[u] + max(0ll, dp2[v] + sum - (ma[u] - 2 * co) - 2 * co);
      dp[u] = dp[v] + co;
    } else {
      dp2[u] = w[u] + max(0ll, dp2[v] + sum - 2 * co);
      dp[u] = dp[v] - co + ma[u];
    }
    dfs2(u, v);
  }
}
int lca(int x, int y) {
  if (dep[x] > dep[y]) swap(x, y);
  if (dep[x] != dep[y]) {
    for (int i = 19; i >= 0; i--) {
      if (dep[fa[i][y]] > dep[x]) y = fa[i][y];
    }
    if (x != y) y = fa[0][y];
  }
  if (x == y) return x;
  for (int i = 19; i >= 0; i--) {
    if (fa[i][y] != fa[i][x]) {
      y = fa[i][y];
      x = fa[i][x];
    }
  }
  return fa[0][x];
}
int main() {
  int n, q;
  scanf("%d %d", &n, &q);
  for (int i = 0; i < n; i++) {
    scanf("%I64d", &w[i]);
  }
  for (int i = 0; i < n - 1; i++) {
    int x, y;
    long long co;
    scanf("%d %d %I64d", &x, &y, &co);
    x--, y--;
    nei[x].push_back(make_pair(y, co));
    nei[y].push_back(make_pair(x, co));
  }
  dfs(0, 0, 0);
  dp[0] = w[0];
  dp2[0] = w[0];
  dfs2(0, -1);
  for (int i = 1; i < 20; i++) {
    for (int j = 0; j < n; j++) {
      fa[i][j] = fa[i - 1][fa[i - 1][j]];
    }
  }
  while (q--) {
    int x, y;
    scanf("%d %d", &x, &y);
    x--, y--;
    printf("%I64d\n", dp[x] + dp[y] - 2 * dp[lca(x, y)] + ma[lca(x, y)] +
                          dp2[lca(x, y)] - w[lca(x, y)]);
  }
}

### Prompt

Construct a cpp code solution to the problem outlined:

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 Pr(p) = ∑_{v ∈ distinct vertices in  p}{a_v} - ∑_{e ∈ distinct edges in  p}{k_e ⋅ 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 Pr(p).

Input

The first line contains two integers n and q (2 ≤ n ≤ 3 ⋅ 10^5, 1 ≤ q ≤ 4 ⋅ 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, ..., a_n (1 ≤ a_i ≤ 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 ≤ u_i, v_i ≤ n, u_i ≠ v_i, 1 ≤ w_i ≤ 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 ≤ qu_i, qv_i ≤ n) — endpoints of the 2-path you need to find.

Output

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

Example

Input

7 6
6 5 5 3 2 1 2
1 2 2
2 3 2
2 4 1
4 5 1
6 4 2
7 3 25
1 1
4 4
5 6
6 4
3 4
3 7


Output

9
9
9
8
12
-14

Note

Explanation of queries: 

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

### Response

```cpp
#include <bits/stdc++.h>
using namespace std;
vector<pair<int, long long> > nei[300005];
int fa[20][300005], dep[300005];
long long w[300005], ma[300005];
void dfs(int v, int pa, int depth) {
  fa[0][v] = pa;
  ma[v] = w[v];
  dep[v] = depth;
  for (int i = 0; i < nei[v].size(); i++) {
    int u = nei[v][i].first;
    long long co = nei[v][i].second;
    if (u == pa) continue;
    dfs(u, v, depth + 1);
    ma[v] = max(ma[v] + ma[u] - 2 * co, ma[v]);
  }
}
long long dp[300005], dp2[300005];
void dfs2(int v, int pa) {
  long long sum = 0;
  for (int i = 0; i < nei[v].size(); i++) {
    int u = nei[v][i].first;
    long long co = nei[v][i].second;
    if (u == pa) continue;
    if (ma[u] - 2 * co > 0) sum += ma[u] - 2 * co;
  }
  for (int i = 0; i < nei[v].size(); i++) {
    int u = nei[v][i].first;
    long long co = nei[v][i].second;
    if (u == pa) continue;
    if (ma[u] - 2 * co > 0) {
      dp2[u] = w[u] + max(0ll, dp2[v] + sum - (ma[u] - 2 * co) - 2 * co);
      dp[u] = dp[v] + co;
    } else {
      dp2[u] = w[u] + max(0ll, dp2[v] + sum - 2 * co);
      dp[u] = dp[v] - co + ma[u];
    }
    dfs2(u, v);
  }
}
int lca(int x, int y) {
  if (dep[x] > dep[y]) swap(x, y);
  if (dep[x] != dep[y]) {
    for (int i = 19; i >= 0; i--) {
      if (dep[fa[i][y]] > dep[x]) y = fa[i][y];
    }
    if (x != y) y = fa[0][y];
  }
  if (x == y) return x;
  for (int i = 19; i >= 0; i--) {
    if (fa[i][y] != fa[i][x]) {
      y = fa[i][y];
      x = fa[i][x];
    }
  }
  return fa[0][x];
}
int main() {
  int n, q;
  scanf("%d %d", &n, &q);
  for (int i = 0; i < n; i++) {
    scanf("%I64d", &w[i]);
  }
  for (int i = 0; i < n - 1; i++) {
    int x, y;
    long long co;
    scanf("%d %d %I64d", &x, &y, &co);
    x--, y--;
    nei[x].push_back(make_pair(y, co));
    nei[y].push_back(make_pair(x, co));
  }
  dfs(0, 0, 0);
  dp[0] = w[0];
  dp2[0] = w[0];
  dfs2(0, -1);
  for (int i = 1; i < 20; i++) {
    for (int j = 0; j < n; j++) {
      fa[i][j] = fa[i - 1][fa[i - 1][j]];
    }
  }
  while (q--) {
    int x, y;
    scanf("%d %d", &x, &y);
    x--, y--;
    printf("%I64d\n", dp[x] + dp[y] - 2 * dp[lca(x, y)] + ma[lca(x, y)] +
                          dp2[lca(x, y)] - w[lca(x, y)]);
  }
}

```