51nod2004 终结之时 (支配树+树剖+树链的并)

link

我永远喜欢洛天依

给定一张图世末积雨云，你需要维护其支配树：

单点修改，子树修改，树链修改

子树求和，树链求和，多条树链的并集求和

撤销之前的操作

可以先用 Lengauer-Tarjan 算法 求出图的支配树，然后把它树剖，开一个线段树维护

至于多条树链的并集求和（有点像虚树的那种操作），现将所有点按照 dfn 排序，然后按照 dfn 的顺序依次加点

枚举节点a[i]，每次假装a[1..i]的答案中除了a[i]到根的路径没有求值之外其它的都求值了，那么转移的时候需要加上当前点到lca的距离。

aaarticlea/png;base64,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" alt="">

如图，我们从1转移到2时候，要加上绿色的链这一段的值

最后转移完后，我们再加上最后一个点到根的距离即可

至于撤销操作，我们可以考虑建立主席树，当然也可以不建主席树暴力撤销，因为每一步操作最多只会被撤销一次。

代码：

#include <cstdio>
#include <vector>
#include <algorithm>
using namespace std;

int n, m;

namespace dominate_tree
{
	vector<int> g[50010], g1[50010], g2[50010];
	int sdom[50010], bel[50010], val[50010], idom[50010];
	int dfn[50010], dfntot, id[50010], fa[50010];

	void dfs(int x)
	{
		dfn[x] = ++dfntot, id[dfntot] = x;
		for (int i : g[x]) if (dfn[i] == 0) fa[i] = x, dfs(i);
	}

	int getf(int x)
	{
		if (x == bel[x]) return x;
		int rt = getf(bel[x]);
		if (dfn[sdom[val[bel[x]]]] < dfn[sdom[val[x]]])
			val[x] = val[bel[x]];
		return bel[x] = rt;
	}

	void work()
	{
		for (int i = 1; i <= n; i++) sdom[i] = bel[i] = val[i] = i;
		dfs(1);
		for (int i = n; i >= 2; i--)
		{
			int x = id[i];
			for (int j : g1[x])
				if (dfn[j])
				{
					getf(j);
					if (dfn[sdom[val[j]]] < dfn[sdom[x]])
						sdom[x] = sdom[val[j]];
				}
			g2[sdom[x]].push_back(x);
			x = bel[x] = fa[x];
			for (int j : g2[x])
			{
				getf(j);
				if (sdom[val[j]] == x) idom[j] = x;
				else idom[j] = val[j];
			}
		}
		for (int i = 2; i <= n; i++)
		{
			int x = id[i];
			if (idom[x] != sdom[x]) idom[x] = idom[idom[x]];
		}
	}
}

vector<int> out[50010];
int fa[50010], depth[50010], wson[50010], weight[50010];
int dfn[50010], top[50010], tot, val[50010], stop, tmp[50010];
long long tree[200010], lazy[200010];
struct ch { int type, u, w; } s[100010];

void pushdown(int x, int cl, int mid, int cr)
{
	lazy[x * 2] += lazy[x], lazy[x * 2 + 1] += lazy[x];
	tree[x * 2] += lazy[x] * (mid - cl + 1);
	tree[x * 2 + 1] += lazy[x] * (cr - mid);
	lazy[x] = 0;
}

void chenge(int x, int cl, int cr, int L, int R, int k)
{
	if (cr < L || R < cl) return;
	if (L <= cl && cr <= R) { lazy[x] += k, tree[x] += (cr - cl + 1) * (long long)k; return; }
	int mid = (cl + cr) / 2;
	pushdown(x, cl, mid, cr);
	chenge(x * 2, cl, mid, L, R, k), chenge(x * 2 + 1, mid + 1, cr, L, R, k);
	tree[x] = tree[x * 2] + tree[x * 2 + 1];
}

long long query(int x, int cl, int cr, int L, int R)
{
	if (cr < L || R < cl) return 0;
	if (L <= cl && cr <= R) return tree[x];
	int mid = (cl + cr) / 2;
	pushdown(x, cl, mid, cr);
	return query(x * 2, cl, mid, L, R) + query(x * 2 + 1, mid + 1, cr, L, R);
}

void dfs1(int x)
{
	wson[x] = -1, weight[x] = 1;
	for (int i : out[x])
	{
		fa[i] = x, depth[i] = depth[x] + 1, dfs1(i), weight[x] += weight[i];
		if (wson[x] == -1 || weight[i] > weight[wson[x]]) wson[x] = i;
	}
}

void dfs2(int x, int topf)
{
	dfn[x] = ++tot, top[x] = topf;
	if (wson[x] != -1)
	{
		dfs2(wson[x], topf);
		for (int i : out[x]) if (i != wson[x]) dfs2(i, i);
	}
}

int lca(int x, int y)
{
	while (top[x] != top[y])
	{
		if (depth[top[x]] < depth[top[y]]) swap(x, y);
		x = fa[top[x]];
	}
	if (depth[x] > depth[y]) swap(x, y);
	return x;
}

long long getans(int x)
{
	long long res = 0;
	while (x)
		res += query(1, 1, n, dfn[top[x]], dfn[x]), x = fa[top[x]];
	return res;
}

int main()
{
	scanf("%d%d", &n, &m);
	for (int i = 1; i <= n; i++) scanf("%d", &val[i]);
	for (int x, y, i = 1; i <= m; i++)
		scanf("%d%d", &x, &y), dominate_tree::g[x].push_back(y), dominate_tree::g1[y].push_back(x);
	dominate_tree::work();
	for (int i = 2; i <= n; i++) out[dominate_tree::idom[i]].push_back(i);
	dfs1(1), dfs2(1, 1);
	for (int i = 1; i <= n; i++) chenge(1, 1, n, dfn[i], dfn[i], val[i]);
	int q; scanf("%d", &q);
	for (int i = 1; i <= q; i++)
	{
		char tp;
		scanf(" %c", &tp);
		if (tp == 'C')
		{
			int opd, u, w;
			scanf("%d%d%d", &opd, &u, &w);
			s[++stop] = (ch){opd, u, w};
			if (opd == 1) chenge(1, 1, n, dfn[u], dfn[u], w);
			if (opd == 2) chenge(1, 1, n, dfn[u], dfn[u] + weight[u] - 1, w);
			if (opd == 3)
				while (u) chenge(1, 1, n, dfn[top[u]], dfn[u], w), u = fa[top[u]];
		}
		else if (tp == 'Q')
		{
			int opd, u;
			scanf("%d%d", &opd, &u);
			if (opd == 1) printf("%lld\n", query(1, 1, n, dfn[u], dfn[u] + weight[u] - 1));
			if (opd == 2) printf("%lld\n", getans(u));
			if (opd == 3)
			{
				long long res = 0;
				for (int j = 1; j <= u; j++) scanf("%d", &tmp[j]);
				sort(tmp + 1, tmp + 1 + u, [](int x, int y) { return dfn[x] < dfn[y]; });
				for (int j = 2; j <= u; j++)
				{
					int lc = lca(tmp[j - 1], tmp[j]);
					if (lc != tmp[j - 1])
						res += getans(tmp[j - 1]) - getans(lc);
				}
				res += getans(tmp[u]);
				printf("%lld\n", res);
			}
		}
		else
		{
			int k;
			scanf("%d", &k);
			while (stop > 0 && k > 0)
			{
				int opd = s[stop].type, u = s[stop].u, w = -s[stop].w;
				if (opd == 1) chenge(1, 1, n, dfn[u], dfn[u], w);
				if (opd == 2) chenge(1, 1, n, dfn[u], dfn[u] + weight[u] - 1, w);
				if (opd == 3)
					while (u) chenge(1, 1, n, dfn[top[u]], dfn[u], w), u = fa[top[u]];
				stop--, k--;
			}
		}
	}
	return 0;
}