BZOJ4912 [Sdoi2017]天才黑客 【虚树 + 最短路】

题目链接

BZOJ4912

题解

转移的代价是存在于边和边之间的

所以把边看做点，跑最短路

但是这样做需要把同一个点的所有入边和所有出边之间连边

\(O(m^2)\)的连边无法接受

需要优化建图

膜一下Claris的方法

对每个点，取出其入边出边，按在字典树上的\(dfs\)序排序

按\(dfs\)序排序，实际上就是将字符串排序了

按照后缀数组的理论，两点之间的\(lcp\)就是两点之间相邻\(lcp\)，也就是\(height\)数组的最小值

对于一个位置的\(height\)，两边的点之间联通所需最小代价不超过\(height\)

所以可以用\(lca\)求出\(height\)数组，建立前缀后缀虚点

以前缀为例，横向边为\(0\)，纵向边为\(height\)的大小



这样如果想从左边的点到达右边的点，就只需经过中间最小的\(height\)了

类似可以建立后缀点处理从右到左的情况

复杂度\(O(mlogm)\)

#include<algorithm>
#include<iostream>
#include<cstdlib>
#include<cstring>
#include<cstdio>
#include<vector>
#include<queue>
#include<cmath>
#include<map>
#define LL long long int
#define REP(i,n) for (int i = 1; i <= (n); i++)
#define Redge(u) for (int k = h[u],to; k; k = ed[k].nxt)
#define cls(s,v) memset(s,v,sizeof(s))
#define mp(a,b) make_pair<int,int>(a,b)
#define cp pair<int,int>
using namespace std;
const int maxn = 500005,maxm = 2000005,INF = 2000000000;
inline int read(){
	int out = 0,flag = 1; char c = getchar();
	while (c < 48 || c > 57){if (c == '-') flag = 0; c = getchar();}
	while (c >= 48 && c <= 57){out = (out << 1) + (out << 3) + c - 48; c = getchar();}
	return flag ? out : -out;
}
struct EDGE{int to,nxt,w;}ed[maxm];
struct Trie{
	int h[maxn],ne,dfn[maxn],fa[maxn][16],dep[maxn],cnt,n;
	EDGE ed[maxm];
	void init(){REP(i,n) h[i] = 0; ne = 1; cnt = 0;}
	void build(int u,int v){ed[++ne] = (EDGE){v,h[u]}; h[u] = ne;}
	void dfs(int u){
		dfn[u] = ++cnt;
		REP(i,15) fa[u][i] = fa[fa[u][i - 1]][i - 1];
		Redge(u) {
			fa[to = ed[k].to][0] = u;
			dep[to] = dep[u] + 1; dfs(to);
		}
	}
	int lca(int u,int v){
		if (dep[u] < dep[v]) swap(u,v);
		for (int D = dep[u] - dep[v],i = 0; (1 << i) <= D; i++)
			if (D & (1 << i)) u = fa[u][i];
		if (u == v) return u;
		for (int i = 15; ~i; i--)
			if (fa[u][i] != fa[v][i]){
				u = fa[u][i];
				v = fa[v][i];
			}
		return fa[u][0];
	}
}T;
int h[maxn],ne = 1;
int n,m,N,w[maxn],pos[maxn];
int c[maxn],ci,height[maxn];
int pu[maxn],pd[maxn],su[maxn],sd[maxn];
int d[maxn],vis[maxn];
vector<int> in[maxn],out[maxn];
priority_queue<cp,vector<cp>,greater<cp> > q;
inline void build(int u,int v,int w){ed[++ne] = (EDGE){v,h[u],w}; h[u] = ne;}
inline bool cmp(const int& a,const int& b){
	return T.dfn[pos[abs(a)]] < T.dfn[pos[abs(b)]];
}
void init(){
	T.init(); ne = 1; N = 0; cls(h,0); cls(w,0);
	REP(i,n) in[i].clear(),out[i].clear();
}
void Build(){
	for (int p = 1; p <= n; p++){
		if (!in[p].size() && !out[p].size()) continue;
		ci = 0;
		for (unsigned int i = 0; i < in[p].size(); i++)
			c[++ci] = in[p][i];
		for (unsigned int i = 0; i < out[p].size(); i++)
			c[++ci] = -out[p][i];
		sort(c + 1,c + 1 + ci,cmp);
		for (int i = 1; i <= ci; i++){
			pu[i] = ++N,pd[i] = ++N;
			su[i] = ++N,sd[i] = ++N;
			if (i > 1){
				build(pu[i - 1],pu[i],0);
				build(pd[i - 1],pd[i],0);
				build(su[i],su[i - 1],0);
				build(sd[i],sd[i - 1],0);
			}
			if (c[i] >= 0) build(c[i],pu[i],0),build(c[i],su[i],0);
			else c[i] *= -1,build(pd[i],c[i],0),build(sd[i],c[i],0);
		}
		for (int i = 1; i < ci; i++){
			int tmp = T.dep[T.lca(pos[c[i]],pos[c[i + 1]])];
			build(pu[i],pd[i + 1],tmp);
			build(su[i + 1],sd[i],tmp);
		}
	}
}
void dijkstra(){
	for (int i = 0; i <= N; i++) d[i] = INF,vis[i] = false;
	for (unsigned int j = 0; j < out[1].size(); j++)
		q.push(mp(d[out[1][j]] = 0,out[1][j]));
	int u;
	while (!q.empty()){
		u = q.top().second; q.pop();
		if (vis[u]) continue;
		vis[u] = true;
		Redge(u) if (!vis[to = ed[k].to] && d[to] > d[u] + ed[k].w + w[u]){
			d[to] = d[u] + ed[k].w + w[u];
			q.push(mp(d[to],to));
		}
	}
}
void print(){
	for (int i = 2; i <= n; i++){
		int ans = INF;
		for (unsigned int j = 0; j < in[i].size(); j++)
			ans = min(ans,d[in[i][j]] + w[in[i][j]]);
		printf("%d\n",ans);
	}
}
int main(){
	int Case = read();
	while (Case--){
		n = read(); m = read(); T.n = read(); init();
		int a,b;
		REP(i,m){
			a = read(); b = read();
			w[i] = read(); pos[i] = read();
			out[a].push_back(i);
			in[b].push_back(i);
		}
		N = m;
		for (int i = 1; i < T.n; i++){
			a = read(); b = read(); read();
			T.build(a,b);
		}
		T.dfs(1);
		Build();
		dijkstra();
		print();
	}
	return 0;
}