P3128 [USACO15DEC]Max Flow P（树上倍增和树链剖分）

思路1（树上倍增$ + $树上差分）

每次都修改一条从\(u\)到\(v\)，不就是树上差分的专门操作吗？？

直接用倍增求\(LCA\)，每次\(d[u]++,d[v]++,d[LCA(u,v)]--,d[f[LCA(u,v)][0]]--\)。

最后记得算下前缀和。

代码1

#include <iostream>
#include <cstring>
using namespace std;
const int N = 50010,M = 2 * N,MAX_LOG = 20;
int n,m;
int h[N],e[M],ne[M],idx;
int w[N];
int dep[N];
int f[N][MAX_LOG];
void add (int a,int b) {
	e[idx] = b;
	ne[idx] = h[a];
	h[a] = idx++;
}
void dfs1 (int u,int fa) {
	f[u][0] = fa;
	for (int i = 1;i <= MAX_LOG - 1;i++) f[u][i] = f[f[u][i - 1]][i - 1];
	for (int i = h[u];~i;i = ne[i]) {
		int j = e[i];
		if (j == fa) continue;
		dep[j] = dep[u] + 1;
		dfs1 (j,u);
	}
}
int get_LCA (int a,int b) {
	if (dep[a] < dep[b]) swap (a,b);
	for (int i = MAX_LOG - 1;i >= 0;i--) {
		if (dep[f[a][i]] >= dep[b]) a = f[a][i];
	}
	if (a == b) return a;
	for (int i = MAX_LOG - 1;i >= 0;i--) {
		if (f[a][i] != f[b][i]) a = f[a][i],b = f[b][i];
	}
	return f[a][0];
}
int dfs2 (int u,int fa) {
	int ans = 0;
	for (int i = h[u];~i;i = ne[i]) {
		int j = e[i];
		if (j == fa) continue;
		ans = max (ans,dfs2 (j,u));
		w[u] += w[j];
	}
	return max (ans,w[u]);
}
int main () {
	memset (h,-1,sizeof (h));
	cin >> n >> m;
	for (int i = 1;i <= n - 1;i++) {
		int a,b;
		cin >> a >> b;
		add (a,b),add (b,a);
	}
	dfs1 (1,0);
	while (m--) {
		int a,b;
		cin >> a >> b;
		int anc = get_LCA (a,b);
		w[a]++,w[b]++,w[anc]--,w[f[anc][0]]--;
	}
	cout << dfs2 (1,0) << endl;
	return 0;
}

思路2（树链剖分）

修改一条\(u\)到\(v\)的路径，查询整棵树的最大值，不就是树剖的模板吗？？

直接套模板（懒得讲

代码2

#include <iostream>
#include <cstring>
using namespace std;
const int N = 50010,M = 2 * N,MAX_LOG = 20;
int n,m;
int h[N],e[M],ne[M],idx;
int timestamp;
int dep[N],s[N],son[N],fa[N];
int id[N],top[N];
struct segment_tree_node {
	int l,r;
	int maxx,add;
}tr[4 * N];
void add (int a,int b) {
	e[idx] = b;
	ne[idx] = h[a];
	h[a] = idx++;
}
void push_up (int u) {
	tr[u].maxx = max (tr[u << 1].maxx,tr[u << 1 | 1].maxx);
}
void push_down (int u) {
	auto &root = tr[u],&left = tr[u << 1],&right = tr[u << 1 | 1];
	if (root.add) {
		left.maxx += root.add,left.add += root.add;
		right.maxx += root.add,right.add += root.add;
		root.add = 0;
	}
}
void build_segment_tree (int u,int l,int r) {
	if (l == r) {
		tr[u] = {l,r,0,0};
		return ;
	}
	tr[u] = {l,r};
	int mid = l + r >> 1;
	build_segment_tree (u << 1,l,mid),build_segment_tree (u << 1 | 1,mid + 1,r);
	push_up (u);
}
void modify (int u,int l,int r,int d) {
	if (l <= tr[u].l && tr[u].r <= r) {
		tr[u].add += d,tr[u].maxx += d;
		return ;
	}
	push_down (u);
	int mid = tr[u].l + tr[u].r >> 1;
	if (l <= mid) modify (u << 1,l,r,d);
	if (r >= mid + 1) modify (u << 1 | 1,l,r,d);
	push_up (u);
}
int query_max (int u,int l,int r) {
	if (l <= tr[u].l && tr[u].r <= r) return tr[u].maxx;
	push_down (u);
	int mid = l + r >> 1;
	int ans = 0;
	if (l <= mid) ans = max (ans,query_max (u << 1,l,r));
	if (r >= mid + 1) ans = max (ans,query_max (u << 1 | 1,l,r));
	return ans;
}
void dfs1 (int u,int f) {
	dep[u] = dep[f] + 1,s[u] = 1,fa[u] = f;
	for (int i = h[u];~i;i = ne[i]) {
		int j = e[i];
		if (j == f) continue;
		dfs1 (j,u);
		s[u] += s[j];
		if (s[j] > s[son[u]]) son[u] = j;
	}
}
void dfs2 (int u,int top_node) {
	id[u] = ++timestamp,top[u] = top_node;
	if (!son[u]) return ;
	dfs2 (son[u],top_node);
	for (int i = h[u];~i;i = ne[i]) {
		int j = e[i];
		if (j == fa[u] || j == son[u]) continue;
		dfs2 (j,j);
	}
}
void modify_path (int a,int b) {
	while (top[a] != top[b]) {
		if (dep[top[a]] < dep[top[b]]) swap (a,b);
		modify (1,id[top[a]],id[a],1);
		a = fa[top[a]];
	}
	if (dep[a] > dep[b]) swap (a,b);
	modify (1,id[a],id[b],1);
}
int query_subtree (int u) {
	return query_max (1,id[u],id[u] + s[u] - 1);
}
int main () {
	memset (h,-1,sizeof (h));
	cin >> n >> m;
	for (int i = 1;i <= n - 1;i++) {
		int a,b;
		cin >> a >> b;
		add (a,b),add (b,a);
	}
	build_segment_tree (1,1,n),dfs1 (1,0),dfs2 (1,1);
	while (m--) {
		int a,b;
		cin >> a >> b;
		modify_path (a,b);
	}
	cout << query_subtree (1) << endl;
	return 0;
}