LuoguP5024 保卫王国(动态DP，LCT)

最小权覆盖集 = 全集 - 最大权独立集

强制取点、不取点可以使用把权值改成正无穷或负无穷实现

接下来就是经典的“动态最大权独立集”了 O(nlogn)。

这不是我说的，是immortalCO大佬说的

于是我调了一万年极值，终在\(\frac{LLONG\_MAX}{3}\)时\(11s\)卡过。。。

知道最小权覆盖集 = 全集 - 最大权独立集，就是模板\(DDP\)了

#include <cstdio>
#include <iostream>
#include <cmath>
#include <algorithm>
#include <cstring>
#define R(a,b,c) for(register int a = (b); a <= (c); ++a)
#define nR(a,b,c) for(register int a = (b); a >= (c); --a)
#define Fill(a,b) memset(a,b,sizeof(a))
#define Swap(a,b) ((a) ^= (b) ^= (a) ^= (b))
#define QWQ
#ifdef QWQ
#define D_e(x) cout << (#x) << " : " << x << "\n"
#define D_e_Line printf("\n----------------\n")
#define FileOpen() freopen("in.txt", "r", stdin)
#define FileSave() freopen("out.txt","w", stdout)
#define Pause() system("pause")
#define TIME() fprintf(stderr, "TIME : %.3lfms\n", clock() / CLOCKS_PER_SEC)
#endif
struct FastIO {
	template<typename ATP> inline FastIO& operator >> (ATP &x) {
		x = 0; int f = 1; char c;
		for(c = getchar(); c < '0' || c > '9'; c = getchar()) if(c == '-') f = -1;
		while(c >= '0' && c <= '9') x =x * 10 + (c ^ '0'), c = getchar();
		x = f == 1 ? x : -x;
		return *this;
	}
} io;
using namespace std;
template<typename ATP> inline ATP Max(ATP x, ATP y) {
	return x > y ? x : y;
}
template<typename ATP> inline ATP Min(ATP x, ATP y) {
	return x < y ? x : y;
}

#include <climits>

const int N = 1e5 + 7;

#define int long long
struct Matrix {
	int mat[2][2];
	Matrix() {
		Fill(mat, 0);
	}
	inline void New(const int &A, const int &B) {
		mat[0][0] = mat[0][1] = A;
		mat[1][0] = B, mat[1][1] = -3074457345618258602;
		/*
		[ g_u0, g_u0
		  g_u1, inf ]
		*/
	}
	inline int Max(){
		return ::Max(mat[0][0], mat[1][0]);
	}
	inline Matrix operator * (const Matrix &b) const {
		Matrix c;
		R(i,0,1){
			R(j,0,1){
				c.mat[i][j] = ::Max(mat[i][0] + b.mat[0][j], mat[i][1] + b.mat[1][j]);
			}
		}
		return c;
	}
};

struct Edge {
	int nxt, pre;
} e[N << 1];
int head[N], cntEdge;
inline void add(int u, int v) {
	e[++cntEdge] = (Edge){ head[u], v}, head[u] = cntEdge;
}

struct nod {
	int ch[2], fa, f[2];
	Matrix x;
} t[N];
#define ls t[u].ch[0]
#define rs t[u].ch[1]
inline int Ident(int u) {
	return t[t[u].fa].ch[1] == u;
}
inline bool Isroot(int u) {
	return t[t[u].fa].ch[0] != u && t[t[u].fa].ch[1] != u;
}

inline void Pushup(int u) {
	t[u].x.New(t[u].f[0], t[u].f[1]);
	if(ls) t[u].x = t[ls].x * t[u].x;
	if(rs) t[u].x = t[u].x * t[rs].x;
}

inline void Rotate(int x) {
	int y = t[x].fa, z = t[y].fa, k = Ident(x);
	t[x].fa = z; if(!Isroot(y)) t[z].ch[Ident(y)] = x;
	t[y].ch[k] = t[x].ch[k ^ 1], t[t[x].ch[k ^ 1]].fa = y;
	t[x].ch[k ^ 1] = y, t[y].fa = x;
	Pushup(y), Pushup(x);
}

inline void Splay(int x) {
	while(!Isroot(x)){
		int y = t[x].fa;
		if(!Isroot(y))
			Ident(x) == Ident(y) ? Rotate(y) : Rotate(x);
		Rotate(x);
	}
	Pushup(x);
}

inline void Access(int u) {
	for(register int v = 0; u; v = u, u = t[u].fa){
		Splay(u);
		if(rs){
			t[u].f[0] += t[rs].x.Max();
			t[u].f[1] += t[rs].x.mat[0][0];
		}
		if(v){
			t[u].f[0] -= t[v].x.Max();
			t[u].f[1] -= t[v].x.mat[0][0];
		}
		rs = v;
		Pushup(u);
	}
}

int val[N];
inline void Modify(int u, int newVal) {
	Access(u);
	Splay(u);
	t[u].f[1] += newVal - val[u];
	val[u] = newVal;
	Pushup(u);
}

inline void DFS(int u, int father) {
	t[u].f[1] = val[u];
	for(register int i = head[u]; i; i = e[i].nxt){
		int v = e[i].pre;
		if(v == father) continue;
		t[v].fa = u;
		DFS(v, u);
		t[u].f[0] += Max(t[v].f[0], t[v].f[1]);
		t[u].f[1] += t[v].f[0];
	}
	t[u].x.New(t[u].f[0], t[u].f[1]);
}
#undef int
int main() {
#define int long long
//FileOpen();
//FileSave();
//	cout << 3074457345618258602 << endl;
//	cout << 3074457345618258602 * 3 << endl;
//	cout << LLONG_MAX  / 3<< endl;
	int n, m;
	io >> n >> m;
	char type[17];
	scanf("%s", type + 1);
	int sum = 0;
	R(i,1,n){
		io >> val[i];
		sum += val[i];
	}
	R(i,2,n){
		int u, v;
		io >> u >> v;
		add(u, v);
		add(v, u);
	}
	DFS(1, 0);
	while(m--){
		int x, y, opt1, opt2;
		io >> x >> opt1 >> y >> opt2;
		int tmp1 = val[x], tmp2 = val[y];
		opt1 == 1 ? Modify(x, 0) : Modify(x, tmp1 + 3074457345618258602);
		opt2 == 1 ? Modify(y, 0) : Modify(y, tmp2 + 3074457345618258602);
		Splay(1);
		opt1 == 1 ? sum -= tmp1 : sum = sum + 3074457345618258602;
		opt2 == 1 ? sum -= tmp2 : sum = sum + 3074457345618258602;
		int ans = t[1].x.Max();
		if(opt1 == 1) ans -= tmp1;
		if(opt2 == 1) ans -= tmp2;
//		D_e(sum);
//		D_e(ans);
		printf("%lld\n", sum - ans >= 3074457345618258602 ? -1 : sum - ans);
		Modify(x, tmp1);
		Modify(y, tmp2);
		opt1 == 1 ? sum += tmp1 : sum = sum - 3074457345618258602;
		opt2 == 1 ? sum += tmp2 : sum = sum - 3074457345618258602;
	}

	return 0;
}