LibreOJ 2955 「NOIP2018」保卫王国

考虑一个朴素的树形 DP：
设 $f_{u,0/1}$ 表示选 / 不选 $u$ 这个点的情况下，以 $u$ 为根子树内的最小点覆盖。
有转移

$$\begin{array}{rl}{f}_{u,0}=& \sum _{(u,v)\in E}{f}_{v,1}\\ f_{u,1}=& a_u+\sum _{(u,v)\in E}min(f_{v,0},f_{v,1})\end{array}$$

对轻重儿子分类讨论，设 $g_{u,0/1}$ 表示不考虑重儿子贡献的 $f_u$。
则

$$\begin{array}{rl}{f}_{u,0}=& g_{u,0}+f_{{\text{son}}_u,1}\\ f_{u,1}=& g_{u,1}+min(f_{{\text{son}}_u,0},f_{{\text{son}}_u,1})\end{array}$$

定义 $\otimes$ 为把乘法换成加法，把加法换成去 $min$ 的矩乘。
把转移写成 $\otimes$ 的形式：

$$\left[\begin{array}{cc}\mathrm{\infty }& g_{u,0}\\ g_{u,1}& g_{u,1}\end{array}\right]\otimes \left[\begin{array}{c}{f}_{{\text{son}}_u,0}\\ f_{{\text{son}}_u,1}\end{array}\right]=\left[\begin{array}{c}{f}_{u,0}\\ f_{u,1}\end{array}\right]$$

树剖维护即可。

代码：

#include <cstdio>
#include <algorithm>
#define ls (p << 1)
#define rs (ls | 1)
using namespace std;
const int N = 1e5;
int n,m;
int a[N + 5];
int to[N * 2 + 5],pre[N * 2 + 5],first[N + 5];
inline void add(int u,int v)
{
    static int tot = 0;
    to[++tot] = v,pre[tot] = first[u],first[u] = tot;
}
struct Matrix
{
    long long a[2][2];
    inline Matrix()
    {
        a[0][0] = a[0][1] = a[1][0] = a[1][1] = 0x3f3f3f3f3f3f3f3f;
    }
    inline Matrix(int)
    {
        a[0][0] = a[1][1] = 0,a[0][1] = a[1][0] = 0x3f3f3f3f3f3f3f3f;
    }
    inline Matrix(long long _0,long long _1)
    {
        a[0][0] = 0x3f3f3f3f3f3f3f3f,a[0][1] = _0,a[1][0] = a[1][1] = _1;
    }
    inline Matrix operator*(const Matrix &o)
    {
        Matrix ret;
        for(register int i = 0;i < 2;++i)
            for(register int j = 0;j < 2;++j)
                for(register int k = 0;k < 2;++k)
                    ret.a[i][j] = min(ret.a[i][j],a[i][k] + o.a[k][j]);
        return ret;
    }
} seg[(N << 2) + 10];
void insert(int x,Matrix k,int p,int tl,int tr)
{
    if(tl == tr)
    {
        seg[p] = k;
        return ;
    }
    int mid = tl + tr >> 1;
    x <= mid ? insert(x,k,ls,tl,mid) : insert(x,k,rs,mid + 1,tr);
    seg[p] = seg[ls] * seg[rs];
}
Matrix query(int l,int r,int p,int tl,int tr)
{
    if(l <= tl && tr <= r)
        return seg[p];
    int mid = tl + tr >> 1;
    Matrix ret(1);
    l <= mid && (ret = ret * query(l,r,ls,tl,mid),1);
    r > mid && (ret = ret * query(l,r,rs,mid + 1,tr),1);
    return ret;
}
int fa[N + 5],sz[N + 5],son[N + 5],top[N + 5],id[N + 5],ed[N + 5];
long long f[N + 5][2],g[N + 5][2];
void dfs1(int p)
{
    sz[p] = 1,f[p][1] = a[p];
    for(register int i = first[p];i;i = pre[i])
        if(to[i] ^ fa[p])
        {
            fa[to[i]] = p,dfs1(to[i]),sz[p] += sz[to[i]];
            if(!son[p] || sz[to[i]] > sz[son[p]])
                son[p] = to[i];
            f[p][0] += f[to[i]][1],f[p][1] += min(f[to[i]][0],f[to[i]][1]);
        }
}
void dfs2(int p)
{
    static int tot = 0;
    id[p] = ++tot,g[p][1] = a[p];
    if(son[p])
        top[son[p]] = top[p],dfs2(son[p]);
    else
        ed[top[p]] = id[p];
    for(register int i = first[p];i;i = pre[i])
        if(!id[to[i]])
            top[to[i]] = to[i],g[p][0] += f[to[i]][1],g[p][1] += min(f[to[i]][0],f[to[i]][1]),dfs2(to[i]);
}
void update(int x)
{
    for(;x;x = fa[x])
    {
        insert(id[x],Matrix(g[x][0],g[x][1]),1,1,n),x = top[x];
        Matrix cur = query(id[x],ed[x],1,1,n);
        g[fa[x]][0] -= f[x][1],g[fa[x]][1] -= min(f[x][0],f[x][1]);
        f[x][0] = cur.a[0][1],f[x][1] = cur.a[1][1];
        g[fa[x]][0] += f[x][1],g[fa[x]][1] += min(f[x][0],f[x][1]);
    }
}
int main()
{
    freopen("defense.in","r",stdin),freopen("defense.out","w",stdout);
    scanf("%d%d%*s",&n,&m);
    for(register int i = 1;i <= n;++i)
        scanf("%d",a + i);
    int u,v;
    for(register int i = 1;i < n;++i)
        scanf("%d%d",&u,&v),add(u,v),add(v,u);
    top[1] = 1,dfs1(1),dfs2(1);
    for(register int i = 1;i <= n;++i)
        insert(id[i],Matrix(g[i][0],g[i][1]),1,1,n);
    int x,y;
    for(long long _0,_1;m;--m)
    {
        scanf("%d%d%d%d",&u,&x,&v,&y),x ^= 1,y ^= 1;
        _0 = g[u][x],_1 = g[v][y];
        g[u][x] = 0x3f3f3f3f3f3f3f3f,update(u);
        g[v][y] = 0x3f3f3f3f3f3f3f3f,update(v);
        printf("%lld\n",min(f[1][0],f[1][1]) < 0x3f3f3f3f3f3f3f3f ? min(f[1][0],f[1][1]) : -1LL);
        g[u][x] = _0,update(u);
        g[v][y] = _1,update(v);
    }
}