BZOJ 2286: [Sdoi2011]消耗战虚树-优快云博客

本文深入探讨了在树形数据结构中，使用最低公共祖先(LCA)算法和动态规划(DP)来解决复杂路径最小值问题的方法。通过具体的C++代码实现，展示了如何构建树的邻接矩阵，进行深度优先搜索，以及如何利用LCA和DP求解从根节点到每个叶子节点的路径上最小边权值。

Code:

#include <bits/stdc++.h>
#define setIO(s) freopen(s".in", "r", stdin) 
#define maxn 500004 
#define LOG 23 
#define inf 100000000000
#define ll long long 
using namespace std; 
vector <int> G[maxn];  
int edges, tim, n, top; 
int hd[maxn], to[maxn << 1], nex[maxn << 1], val[maxn << 1];  
int dfn[maxn], f[LOG][maxn], arr[maxn], S[maxn], mk[maxn], dep[maxn]; 
ll mn[maxn];  
inline void addedge(int u, int v, int c) 
{
	nex[++edges] = hd[u], hd[u] = edges, to[edges] = v, val[edges] = c; 
}
void dfs1(int u, int ff) 
{ 
	f[0][u] = ff; 
	for(int i = 1; i < 22; ++i) f[i][u] = f[i - 1][f[i - 1][u]]; 
	dep[u] = dep[ff] + 1, dfn[u] = ++tim; 
    for(int i = hd[u]; i ; i = nex[i])
    {
    	int v = to[i]; 
    	if(v == ff) continue; 
    	mn[v] = min(mn[u], 1ll*val[i]); 
    	dfs1(v, u); 
    }
}
inline int LCA(int a, int b)
{
	if(dep[a] > dep[b]) swap(a, b); 
	if(dep[a] != dep[b])
	{
		for(int i = 21; i >= 0; --i) if(dep[f[i][b]] >= dep[a]) b = f[i][b];  
	}
    if(a == b) return a; 
    for(int i = 21; i >= 0; --i) if(f[i][a] != f[i][b]) a = f[i][a], b = f[i][b]; 
    return f[0][a];  
} 
bool cmp(int a, int b)
{
	return dfn[a] < dfn[b];  
}
inline void add_edge(int u, int v)
{ 
	G[u].push_back(v); 
}
inline void insert(int x)
{
	if(top <= 1) 
	{
		S[++top] = x; 
		return; 
	}
	int lca = LCA(x, S[top]); 
	if(lca == S[top]) return; 
	while(top > 1 && dep[S[top - 1]] >= dep[lca]) add_edge(S[top - 1], S[top]), --top; 
	if(S[top] != lca) add_edge(lca, S[top]), S[top] = lca; 
	S[++top] = x; 
} 
ll DP(int x)
{ 
	ll sum = 0, re; 
	for(int i = 0; i < G[x].size(); ++i)  sum += DP(G[x][i]);   
	if(mk[x]) re = mn[x]; 
    else re = min(mn[x], sum); 
    mk[x] = 0; 
    G[x].clear(); 
    return re; 
}
int main() 
{ 
	// setIO("input"); 
	scanf("%d",&n); 
	for(int i = 1; i < n ; ++i) 
	{
		int a, b, c; 
		scanf("%d%d%d",&a,&b,&c), addedge(a, b, c), addedge(b, a, c); 
	} 
	dep[1] = 1, mn[1] = inf, dfs1(1, 0);      
	int Q; 
	scanf("%d",&Q); 
	while(Q--)
	{
		int k; 
		scanf("%d",&k);   
		for(int i = 1; i <= k ; ++i) scanf("%d",&arr[i]);      
		sort(arr + 1, arr + 1 + k, cmp);    
	    S[++top] = 1;  
	    for(int i = 1; i <= k ; ++i) insert(arr[i]), mk[arr[i]] = 1;         
	    while(top > 0) add_edge(S[top - 1], S[top]) , --top; 
	    printf("%lld\n",DP(1));  
	    for(int i = 1; i <= k ; ++i) mk[arr[i]] = 0; 
	}
	return 0; 
}