【LG3783】[SDOI2017]天才黑客

【LG3783】[SDOI2017]天才黑客

题面

洛谷

题解

首先我们有一个非常显然的\(O(m^2)\)算法，就是将每条边看成点，

然后将每个点的所有入边和出边暴力连边跑最短路，我们想办法优化这里的连边。

具体怎么做呢，我们将所有入边和出边在\(\text{Trie}\)树上所对应的点放在一起按\(dfs\)序排一遍序，那么相邻两个点的距离就是\(dep_{lca}\)，任意两点之间距离就是他们之间所有的\(dep_{lca}\)取个\(\min\)。

那么如何优化连边呢，我们考虑建如图所示的四排点：

其中\(p\)号节点从\(dfs\)序小的往大的连\(0\)边，\(q\)号点反之。

然后相邻的\(p\)和\(p'\)之间连他们两两之间的\(dep_{lca}\)，\(q\)点亦然。

然后入点向编号对应的\(p,q\)连\(0\)边，\(p',q'\)向出点连\(0\)边，然后发现两点之间的距离都可以取\(\min\)啦，这样子我们就可以直接跑\(dijkstra\)即可。

(具体实现详见代码)

代码

#include <iostream>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <cmath>
#include <algorithm>
#include <vector>
#include <queue>
using namespace std;
inline int gi() {
    register int data = 0, w = 1;
    register char ch = 0;
    while (!isdigit(ch) && ch != '-') ch = getchar();
    if (ch == '-') w = -1, ch = getchar();
    while (isdigit(ch)) data = 10 * data + ch - '0', ch = getchar();
    return w * data;
}
const int INF = 2e9;
const int MAX_N = 1e6 + 5;
typedef vector<int> :: iterator iter;
vector<int> in[MAX_N], ot[MAX_N];
struct Graph { int to, cost, next; } e[MAX_N << 1];
int fir[MAX_N], e_cnt;
void clearGraph() { memset(fir, -1, sizeof(fir)); e_cnt = 0; }
void Add_Edge(int u, int v, int w) { e[e_cnt] = (Graph){v, w, fir[u]}, fir[u] = e_cnt++; }
int pa[16][MAX_N], dep[MAX_N], dfn[MAX_N], tim;
void dfs(int x, int fa) {
	dfn[x] = ++tim;
	if (fa) dep[x] = dep[fa] + 1;
	pa[0][x] = fa;
	for (int i = 1; i < 16; i++)
		pa[i][x] = pa[i - 1][pa[i - 1][x]];
	for (int i = fir[x]; ~i; i = e[i].next) dfs(e[i].to, x);
}
int LCA(int x, int y) {
	if (dep[x] < dep[y]) swap(x, y);
	for (int i = 15; ~i; i--)
		if (dep[pa[i][x]] >= dep[y]) x = pa[i][x];
	if (x == y) return x;
	for (int i = 15; ~i; i--)
		if (pa[i][x] != pa[i][y]) x = pa[i][x], y = pa[i][y];
	return pa[0][x];
}
int N, M, K, tot, v[MAX_N], d[MAX_N];
int t[MAX_N], cnt;
int sl[MAX_N], sr[MAX_N], pl[MAX_N], pr[MAX_N];
bool cmp(const int &i, const int &j) { return dfn[d[abs(i)]] < dfn[d[abs(j)]]; }
void build(int x) {
	cnt = 0;
	for (iter i = in[x].begin(); i != in[x].end(); ++i) t[++cnt] = *i;
	for (iter i = ot[x].begin(); i != ot[x].end(); ++i) t[++cnt] = -*i;
	sort(&t[1], &t[cnt + 1], cmp);
	for (int i = 1; i <= cnt; i++) {
		pl[i] = ++tot, pr[i] = ++tot;
		sl[i] = ++tot, sr[i] = ++tot;
		if (i > 1) {
			Add_Edge(pl[i - 1], pl[i], 0), Add_Edge(pr[i - 1], pr[i], 0);
			Add_Edge(sl[i], sl[i - 1], 0), Add_Edge(sr[i], sr[i - 1], 0);
		}
		if (t[i] > 0) Add_Edge(t[i], pl[i], 0), Add_Edge(t[i], sl[i], 0);
		else t[i] = -t[i], Add_Edge(pr[i], t[i], 0), Add_Edge(sr[i], t[i], 0);
	}
	for (int i = 1; i < cnt; i++) {
		int w = dep[LCA(d[t[i]], d[t[i + 1]])];
		Add_Edge(pl[i], pr[i + 1], w), Add_Edge(sl[i + 1], sr[i], w);
	}
}
priority_queue<pair<int, int>, vector<pair<int, int> >, greater<pair<int, int> > > que;
bool vis[MAX_N];
int dis[MAX_N];
void dijkstra() {
	while (!que.empty()) {
		pair<int, int> p = que.top(); que.pop();
		int x = p.second;
		if (dis[x] < p.first) continue;
		for (int i = fir[x]; ~i; i = e[i].next) {
			int v = e[i].to, w = e[i].cost + ::v[v];
			if (!vis[v] && dis[x] + w < dis[v]) {
				dis[v] = dis[x] + w;
				que.push(make_pair(dis[v], v));
			}
		}
	}
}
int main () {
#ifndef ONLINE_JUDGE
    freopen("cpp.in", "r", stdin);
	freopen("cpp.out", "w", stdout);
#endif
	int T = gi();
	while (T--) {
		clearGraph();
		for (int i = 0; i <= 1e6; i++) v[i] = d[i] = 0, dis[i] = INF, in[i].clear(), ot[i].clear();
		N = gi(), M = tot = gi(), K = gi();
		for (int i = 1; i <= M; i++) {
			int x = gi(), y = gi(); v[i] = gi(), d[i] = gi();
			if (x == 1) que.push(make_pair(dis[i] = v[i], i));
			in[y].push_back(i), ot[x].push_back(i);
		}
		for (int i = 1; i < K; i++) {
			int x = gi(), y = gi(); gi();
			Add_Edge(x, y, 0);
		}
		tim = 0, dfs(1, 0);
		clearGraph();
		for (int i = 1; i <= N; i++) build(i);
		dijkstra();
		for (int i = 2; i <= N; i++) {
			int ans = INF;
			for (iter j = in[i].begin(); j != in[i].end(); ++j) ans = min(ans, dis[*j]);
			printf("%d\n", ans);
		}
	}
    return 0;
}