2019-2020 ICPC Asia Hong Kong Regional Contest --- C. Constructing Ranches

题意：一棵树上每个节点有权值\(a_i\)，问你有多少路径满足以下条件：路径经过的每个点权作为边长，可以构成一个多边形。

解法：

三条边能构成三角形的条件是两边之和大于第三边，四边形是三边之和大于第四边，以此类推。转化一下条件就是\(sum>2*max\)
考虑点分治。对于每个选定的重心，我们假设\(sum_i\)表示i到重心的路径上的点权和，\(mx_i\)表示i到重心路径上最大点权。
那么经过重心x符合条件的一条路径就有\(sum_u+sum_v-a_x>2*max(mx_u,mx_v)\)
那么我们对当前处理的所有结点按照\(mx_i\)排序一下，另外一个结点的sum就是定值了，用树状数组可以维护个数。

#include <bits/stdc++.h>
#define ll long long
using namespace std;

const int maxn = 2e5;

ll ans;
int root,minn,all;
bool vis[maxn + 11] = {false};
vector <pair<ll,ll> > v;
vector <ll> s;
int a[maxn + 11],siz[maxn + 11],f[maxn + 11];
int bit[maxn + 11];
vector <int> edge[maxn + 11];

int lowbit(int x) { return x & (-x); }
int query(int x) { int ans = 0; for (; x; x -= lowbit(x)) ans += bit[x]; return ans; }
void update(int x,int val,int n) { for (; x <= n; x += lowbit(x)) bit[x] += val; }

void dfs(int x,int fa) {
    siz[x] = 1; f[x] = 0;
    for (auto v : edge[x]) {
        if (v == fa || vis[v]) continue;
        dfs(v , x);
        siz[x] += siz[v];
        f[x] = max(f[x] , siz[v]);
    }
    f[x] = max(f[x] , all - siz[x]);
    if (f[x] < minn) { minn = f[x]; root = x; }
}

void find(int x,int fa,ll sum,int mx) {
    v.push_back({max(mx , a[x]) , sum + a[x]});
    for (auto v : edge[x]) {
        if (v == fa || vis[v]) continue;
        find(v , x , sum + a[x] , max(mx , a[x]));
    }
}

ll calc(int x,ll val) {
    find(x , 0 , val , val);
    sort(v.begin() , v.end());
    ll cnt = 0;
    ll add = val ? val : a[x];
    for (auto pi : v) s.push_back(pi.second);
    sort(s.begin() , s.end());
    s.erase(unique(s.begin() , s.end()) , s.end());
    for (int i = 0; i < v.size(); i++) {
        pair <ll,ll> pi = v[i];
        int pos = upper_bound(s.begin() , s.end() , 2 * pi.first + add - pi.second) - s.begin();
        cnt += i - query(pos);
        pos = lower_bound(s.begin() , s.end() , pi.second) - s.begin() + 1;
        update(pos , 1 , s.size());
    }
    for (auto pi : v) {
        int pos = lower_bound(s.begin() , s.end() , pi.second) - s.begin() + 1;
        update(pos , -1 , s.size());
    }
    return cnt;
}

void solve(int x) {
    vis[x] = true;
    ans += calc(x , 0); v.clear(); s.clear();
    for (auto V : edge[x]) {
        if (vis[V]) continue;
        ans -= calc(V , a[x]); v.clear(); s.clear();
    }
    for (auto v : edge[x]) {
        if (vis[v]) continue;
        minn = all = siz[v]; root = 0;
        dfs(v , x);
        solve(root);
    }
}

int main(){
    int t;
    scanf("%d" , &t);
    while (t--) {
        int n;
        scanf("%d" , &n);
        for (int i = 1; i <= n; i++) scanf("%d" , &a[i]);
        for (int i = 1; i < n; i++) {
            int u,v;
            scanf("%d %d" , &u,&v);
            edge[u].push_back(v);
            edge[v].push_back(u);
        }
        root = 0; minn = all = n; ans = 0;
        dfs(1 , 0);
        solve(root);
        printf("%lld\n" , ans);
        for (int i = 1; i <= n; i++) siz[i] = f[i] = 0,vis[i] = false,edge[i].clear();
    }
}