[蓝桥杯]真题讲解：砍树（DFS遍历、图的存储、树上差分与LCA）

一、视频讲解

视频讲解

二、暴力代码

#include
#define int long long
using namespace std;
const int N = 1e5 + 10;

typedef pair<int,int> pii;

vector<int>edge[N];

int n, m;

int w[N];//从每一个边的边权。

map<pii, int>id;//存边的编号

//s是路径的起点，v是路径的重终点，u表示你当前走到了哪个点。
bool dfs(int s, int u, int father, int v)
{
	if(u == v)
	{
		return true;
	}


	for(int i = 0; i < edge[u].size(); i ++)
	{
		int son = edge[u][i];

		if(son == father)
			continue;

		if(dfs(s, son, u, v))
		{
			int ID = id[{u, son}];
			w[ID] ++;
			return true;
		}
	}

	return false;
}

void solve()
{
	cin >> n >> m;
	for(int i = 0; i < n - 1; i ++)
	{
		int x, y; cin >> x >> y;

		edge[x].push_back(y);
		edge[y].push_back(x);
		
		id[{x, y}] = id[{y, x}] = i;
	}

	for(int i = 0; i < m; i ++)
	{
		int x, y; cin >> x >> y;
		dfs(x, x, -1, y);
	}

	int ans = -1;
	for(int i = n - 1; i >= 0; i --)
	{
		if(w[i] == m)
		{
			ans = i + 1;
			break;
		}
	}

	cout << ans << endl;
}

signed main()
{
	ios::sync_with_stdio(0);
	cin.tie(0);
	cout.tie(0);
	int t = 1;
	// cin >> t;
	while(t--)
	solve();
}

三、正解代码

//砍树：树上差分 + 最近公共祖先
#include
using namespace std;
typedef pair<int, int> pii;
const int N = 1e5 + 10;
int n, m;
int w[N];//记录每个点的点权。

map<pii,int>id;
vector<int>edge[N];

int siz[N], dep[N], fa[N], son[N], top[N];

void dfs1(int u, int father)
{
	siz[u] = 1, dep[u] = dep[father] + 1;
	fa[u] = father;
	for(int i = 0; i < edge[u].size(); i ++)
	{
		int s = edge[u][i];
		if(s == fa[u])
			continue;
		dfs1(s, u);
		siz[u] += siz[s];
		if(siz[son[u]] < siz[s])
			son[u] = s;
	}
}

void dfs2(int u, int t)
{
	top[u] = t;
	if(son[u] == 0)
		return;
	dfs2(son[u], t);
	for(int i = 0; i < edge[u].size(); i ++)
	{
		int s = edge[u][i];
		if(s == fa[u] || s == son[u])
			continue;
		dfs2(s, s);
	}
}

int lca(int x, int y)
{
	while(top[x] != top[y])
	{
		if(dep[top[x]] < dep[top[y]])
			swap(x, y);

		x = fa[dep[x]];
	}
	return dep[x] < dep[y] ? x : y;
}

void cal_sum(int u, int father)
{
	for(int i = 0; i < edge[u].size(); i ++)
	{
		int son = edge[u][i];
		if(son == father)
			continue;
		cal_sum(son, u);
		w[u] += w[son];
	}
}

void solve()
{
	 cin >> n >> m;
	 for(int i = 1; i <= n - 1; i ++)
	 {
	 	int x, y; cin >> x >> y;
	 	edge[x].push_back(y);
	 	edge[y].push_back(x);
	 	id[{x, y}] = i;
	 	id[{y, x}] = i;
	 }	

	 dfs1(1, 0);
	 dfs2(1, 1);

	 for(int i = 0; i < m; i ++)
	 {
	 	int a, b; cin >> a >> b;

	 	//做树上差分
	 	w[a] ++, w[b] ++;
	 	w[lca(a, b)] -= 2;
	 }


	 cal_sum(1, 0);


	 int ans = -1;
	 for(int i = 1; i <= n; i ++)
	 {
	 	if(w[i] == m)
	 	{
	 		int ID = id[{i, fa[i]}];
	 		ans = max(ans, ID);
	 	}
	 }
	 cout << ans << endl;
}

signed main()
{
	ios::sync_with_stdio(0);
	cin.tie(0);
	int t = 1;
	// cin >> t;
	while(t--)
	solve();
}