【SDOI2017】天才黑客
题面
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。
具体怎么做内，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。
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; }