JZOJ 4043. 【雅礼集训2015Kzf】洪水

题目

题解

很明显的 \(dp\)

\(f_u = \min(a_u, \sum_{(u,v) \in E}f_v)\)

然后套路的设 \(g_u\) 表示不管重儿子的 \(f_u\)

\(f_u = \min(a_u, g_u+f_{\text{son}_v})\)

然后推一波矩阵

\[\begin{bmatrix}
g_u&a_u\\
\infty&0
\end{bmatrix}\otimes
\begin{bmatrix}
f_{\text{son}_u} \\
0
\end{bmatrix}=
\begin{bmatrix}
f_u\\
0
\end{bmatrix}
\]

\(Code\)

#include<cstdio>
#include<iostream>
#define LL long long
#define ls (p << 1)
#define rs (ls | 1)
using namespace std;

const int N = 2e5 + 5;
const LL INF = 0x3f3f3f3f3f3f3f3f;
int n, m, a[N], h[N], tot;
LL f[N], g[N];

struct edge{int to, nxt;}e[N << 1];
inline void add(int x, int y){e[++tot] = edge{y, h[x]}, h[x] = tot;}

int siz[N], fa[N], son[N], top[N], dfn[N], bot[N], dfc;
void dfs1(int x)
{
	siz[x] = 1;
	for(register int i = h[x]; i; i = e[i].nxt)
	{
		int v = e[i].to;
		if (v == fa[x]) continue;
		fa[v] = x, dfs1(v), siz[x] += siz[v], f[x] += f[v];
		if (siz[v] > siz[son[x]]) son[x] = v;
	}
	if (siz[x] == 1) f[x] = a[x];
	else f[x] = min(f[x], 1LL * a[x]);
}
void dfs2(int x)
{
	dfn[x] = ++dfc;
	if (son[x]) top[son[x]] = top[x], dfs2(son[x]);
	else bot[top[x]] = dfn[x], g[x] = INF;
	for(register int i = h[x]; i; i = e[i].nxt)
	{
		int v = e[i].to;
		if (v == son[x] || v == fa[x]) continue;
		top[v] = v, g[x] += f[v], dfs2(v);
	}
}

struct Matrix{
	LL a[2][2];
	inline Matrix(){a[0][0] = a[0][1] = a[1][0] = a[1][1] = INF;}
	inline Matrix operator*(const Matrix &b)
	{
		Matrix c;
		for(register int i = 0; i < 2; i++)
			for(register int j = 0; j < 2; j++)
				for(register int k = 0; k < 2; k++)
				c.a[i][j] = min(c.a[i][j], a[i][k] + b.a[k][j]);
		return c;
	}
}seg[N << 2];
struct Matrix Node(LL _g, LL _a)
{
	Matrix c;
	c.a[0][0] = _g, c.a[0][1] = _a, c.a[1][0] = INF, c.a[1][1] = 0;
	return c;
}
struct Matrix I()
{
	Matrix c;
	c.a[0][0] = c.a[1][1] = 0, c.a[0][1] = c.a[1][0] = INF;
	return c;
}

void update(int p, int l, int r, int x, Matrix v)
{
	if (l == r) return void(seg[p] = v);
	int mid = (l + r) >> 1;
	if (x <= mid) update(ls, l, mid, x, v);
	else update(rs, mid + 1, r , x, v);
	seg[p] = seg[ls] * seg[rs];
}
Matrix query(int p, int l, int r, int tl, int tr)
{
	if (tl <= l && r <= tr) return seg[p];
	int mid = (l + r) >> 1;
	Matrix ret = I();
	if (tl <= mid) ret = query(ls, l, mid, tl, tr);
	if (tr > mid) ret = ret * query(rs, mid + 1, r, tl, tr);
	return ret;
}

inline void change(int x)
{
	while (x)
	{
		update(1, 1, n, dfn[x], Node(g[x], a[x])), x = top[x];
		Matrix ret = query(1, 1, n, dfn[x], bot[x]);
		g[fa[x]] -= f[x], f[x] = min(ret.a[0][0], ret.a[0][1]), g[fa[x]] += f[x];
		x = fa[x];
	}
}

int main()
{
	scanf("%d", &n);
	for(register int i = 1; i <= n; i++) scanf("%d", a + i);
	for(register int i = 1, u, v; i < n; i++) scanf("%d%d", &u, &v), add(u, v), add(v, u);
	dfs1(1), top[1] = 1, dfs2(1);
	for(register int i = 1; i <= n; i++) update(1, 1, n, dfn[i], Node(g[i], a[i]));
	scanf("%d", &m);
	char op[5];
	for(int x, to; m; --m)
	{
		scanf("%s%d", op, &x);
		if (op[0] == 'Q')
		{
			Matrix ret = query(1, 1, n, dfn[x], bot[top[x]]);
			printf("%lld\n", min(ret.a[0][0], ret.a[0][1]));
		}
		else scanf("%d", &to), a[x] += to, change(x);
	}
}