BZOJ2725 [Violet 6]故乡的梦 【dijkstra+线段树】
【题解】
先跑一遍$S-T$的最短路,记录下路径
预处理出来$dS_{i}$表示$S$到$i$的最短路,$frS_{i}$表示$S$到$i$的最短路上,第一个离开$S-T$最短路的点,同理$dT_{i}$表示$i$到$T$的最短路,$frT_{i}$表示$i$到$T$最短路上,第一个进入$S-T$最短路的点。
然后我们枚举每条边$(x,y)$,那么如果要经过$(x,y)$,那么最短路就是$dS_{x} + w_{x,y} + dT_{y}$,然后我们就可以用一棵线段树来维护这个玩意儿。
# include <stdio.h> # include <algorithm> # include <string.h> # include <stdlib.h> # include <map> # include <queue> # define MP make_pair # define fi first # define se second using namespace std; typedef long long ll; #define pii pair<int,int> #define pil pair<ll,int> int n, m; int head[200010], to[400010], next[400010], w[400010], tot=0; ll dS[200010], dT[200010]; int S, T, from[400010], frS[400010], frT[400010]; bool vis[200010]; int route[200010], rtn = 0, rtpos[200010]; const ll INF = 10000000000000000LL; map< pil, int> bh; int U[200010], V[200010]; ll W[200010]; int Qn, Qs, Qt; inline ll MIN(ll x, ll y) { return x<y?x:y; } inline void add(int u, int v, int tw) { ++tot; next[tot]=head[u]; head[u]=tot; to[tot]=v; w[tot]=tw; } priority_queue<pii, vector< pil >, greater< pil > > Q; inline void dijkstra() { while(!Q.empty()) Q.pop(); int start = S; for (int i=1; i<=n; ++i) dS[i] = INF, vis[i] = 0; dS[start] = 0; Q.push(MP(0, start)); while(!Q.empty()) { pil s = Q.top(); Q.pop(); if(vis[s.se]) continue; vis[s.se] = 1; for (int i=head[s.se]; i; i=next[i]) if(!vis[to[i]] && dS[s.se] + w[i] < dS[to[i]]) { dS[to[i]] = dS[s.se] + w[i]; from[to[i]] = s.se; Q.push(MP(dS[to[i]], to[i])); } } } inline void dijkstra1() { while(!Q.empty()) Q.pop(); int start = S; for (int i=1; i<=n; ++i) dS[i] = INF, vis[i] = 0; dS[start] = 0; Q.push(MP(0, start)); while(!Q.empty()) { pil s = Q.top(); Q.pop(); if(vis[s.se]) continue; if(rtpos[s.se]) frS[s.se] = s.se; vis[s.se] = 1; for (int i=head[s.se]; i; i=next[i]) if(!vis[to[i]] && dS[s.se] + w[i] < dS[to[i]]) { dS[to[i]] = dS[s.se] + w[i]; frS[to[i]] = frS[s.se]; Q.push(MP(dS[to[i]], to[i])); } } } inline void dijkstra2() { while(!Q.empty()) Q.pop(); int start = T; for (int i=1; i<=n; ++i) dT[i] = INF, vis[i] = 0; dT[start] = 0; Q.push(MP(0, start)); while(!Q.empty()) { pil s = Q.top(); Q.pop(); if(vis[s.se]) continue; if(rtpos[s.se]) frT[s.se] = s.se; vis[s.se] = 1; for (int i=head[s.se]; i; i=next[i]) if(!vis[to[i]] && dT[s.se] + w[i] < dT[to[i]]) { dT[to[i]] = dT[s.se] + w[i]; frT[to[i]] = frT[s.se]; Q.push(MP(dT[to[i]], to[i])); } } } //=====================================================// int lc[1200010], rc[1200010]; ll we[1200010]; inline void build(int x, int l, int r) { lc[x] = l, rc[x]= r; we[x]=INF; if(l == r) return; int mid=l+r>>1; build(x<<1, l, mid); build(x<<1|1, mid+1, r); } inline void change(int x, int l, int r, long long val) { int L=lc[x], R=rc[x]; if(l<=L && R<=r) { // printf("insert?! %I64d\n", val); we[x] = MIN(we[x], val); return; } int mid=L+R>>1; if(r<=mid) change(x<<1, l, r, val); else if(l>mid) change(x<<1|1, l, r, val); else { change(x<<1, l, mid, val); change(x<<1|1, mid+1, r, val); } } inline ll query(int x, int pos) { int L=lc[x], R=rc[x]; if(L==R) return we[x]; int mid=L+R>>1; if(pos<=mid) return MIN(we[x], query(x<<1, pos)); else return MIN(we[x], query(x<<1|1, pos)); } //=====================================================// int main() { scanf("%d%d", &n, &m); for (int i=1; i<=m; ++i) { scanf("%d%d%d", &U[i], &V[i], &W[i]); if(U[i]>V[i]) swap(U[i], V[i]); add(U[i], V[i], W[i]); add(V[i], U[i], W[i]); } scanf("%d%d", &S, &T); dijkstra(); for (int i=T; i!=S; i=from[i]) route[++rtn] = i; route[++rtn] = S; // for (int i=1; i<=rtn; ++i) printf("%d --> ", route[i]); // puts(""); // for (int i=rtn; i>=1; --i) printf("%d ", route[i]); // printf("%d\n", dS[T]); for (int i=1; i<=rtn; ++i) rtpos[route[i]] = rtn-i+1; for (int i=2; i<=rtn; ++i) { int posx, posy; posx = min(route[i], route[i-1]); posy = route[i]+route[i-1]-posx; bh[MP(posx, posy)] = rtn-i+1; // printf("relabel: x=%d, y=%d. rei=%d\n", posx, posy, rtn-i+1); } dijkstra1(); dijkstra2(); // printf("@@@@@@@@@@@@@@@@@@@@@@@@@@@\n"); // for (int i=1; i<=n; ++i) printf("%d ", frS[i]); // printf("\n"); // for (int i=1; i<=n; ++i) printf("%d ", frT[i]); // printf("\n"); // for (int i=1; i<=n; ++i) printf("%I64d ", dS[i]); // printf("\n"); // for (int i=1; i<=n; ++i) printf("%I64d ", dT[i]); // printf("\n"); // printf("RTPOS\n"); // for (int i=1; i<=n; ++i) printf("%d ", rtpos[i]); // printf("@@@@@@@@@@@@@@@@@@@@@@@@@@@\n"); build(1, 1, rtn-1); for (int i=1; i<=m; ++i) { if(bh.find(MP(U[i], V[i])) != bh.end()) continue; int left = rtpos[frS[U[i]]], right = rtpos[frT[V[i]]] - 1; // printf(" U[i] = %d, V[i] = %d\n", U[i], V[i]); // printf("checking left=%d right=%d\n", left, right); if(left <= right) { // printf("checking ok!!! we = %I64d\n", dS[U[i]] + dT[V[i]] + W[i]); change(1, left, right, dS[U[i]] + dT[V[i]] + W[i]); } swap(U[i], V[i]); left = rtpos[frS[U[i]]], right = rtpos[frT[V[i]]] - 1; // printf(" U[i] = %d, V[i] = %d\n", U[i], V[i]); // printf("checking left=%d right=%d\n", left, right); if(left <= right) { // printf("checking ok!!! we = %I64d\n", dS[U[i]] + dT[V[i]] + W[i]); change(1, left, right, dS[U[i]] + dT[V[i]] + W[i]); } } // for (int i=1; i<=rtn*4; ++i) // printf("i=%d, leftchild = %d, rightchild = %d, value = %I64d\n", i, lc[i], rc[i], we[i]); scanf("%d", &Qn); while(Qn--) { ll ans; scanf("%d%d", &Qs, &Qt); if(Qs > Qt) swap(Qs, Qt); if(bh.find(MP(Qs, Qt)) == bh.end()) ans = dS[T]; else { int pos = bh[MP(Qs, Qt)]; ans = query(1, pos); } if(ans == INF) puts("Infinity"); else printf("%lld\n", ans); } return 0; }
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