[BZOJ 1576] [Usaco2009 Jan] 安全路经Travel 【树链剖分】

题目链接： BZOJ - 1576

 

题目分析

首先Orz Hzwer的题解。

先使用 dijikstra 求出最短路径树。

那么对于一条不在最短路径树上的边 (u -> v, w) 我们可以先沿树边从 1 走到 u ，再走这条边到 v ，然后再沿树边向上，可以走到 (LCA(u, v), v] 的所有点 （不包括LCA(u, v)！！）。

对于一个属于 (LCA(u, v), v] 的点 x，这种走法的距离为 d[u] + w + d[v] - d[x] ，那么我们就可以用 d[u] + w + d[v] 更新 (LCA(u, v), v] 这一段点的权值，使用树链剖分 + 线段树。

枚举每一条非树边进行更新。

最后每个点 x 的答案就是 x 的权值 - d[x] 。

注意！LCA(u, v) 是不能被这条边更新的！

 

代码

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#include <iostream> #include <cstdio> #include <cstring> #include <cmath> #include <cstdlib> #include <algorithm> #include <queue>   using namespace std;   const int MaxN = 100000 + 5, MaxM = 200000 + 5, MaxLog = 20, INF = 999999999;   int n, m, Index; int Father[MaxN], Depth[MaxN], Top[MaxN], Size[MaxN], Son[MaxN], Pos[MaxN]; int d[MaxN], D[MaxN * 4], Jump[MaxN][MaxLog + 3];   struct Edge {     int u, v, w;     bool Mark;     Edge *Next; } E[MaxM * 2], *P = E, *Pre[MaxN], *Point[MaxN];   inline void AddEdge(int x, int y, int z) {     ++P; P -> u = x; P -> v = y; P -> w = z; P -> Mark = false;     P -> Next = Point[x]; Point[x] = P; }   struct ES {     int x, y;     ES() {}     ES(int a, int b) {         x = a; y = b;     } };   struct Cmp {     bool operator () (ES a, ES b) {         return a.y > b.y;     } };   priority_queue<ES, vector<ES>, Cmp> Q;   bool Visit[MaxN];   void Dijkstra() {     while (!Q.empty()) Q.pop();     for (int i = 1; i <= n; ++i) {         d[i] = INF; Visit[i] = false;     }     d[1] = 0;     for (int i = 1; i <= n; ++i) Q.push(ES(i, d[i]));     ES Now;     int x;     while (!Q.empty()) {         Now = Q.top(); Q.pop();         x = Now.x;         if (Visit[x]) continue;         Visit[x] = true;         for (Edge *j = Point[x]; j; j = j -> Next) {             if (d[x] + (j -> w) < d[j -> v]) {                 d[j -> v] = d[x] + j -> w;                 if (Pre[j -> v] != NULL) Pre[j -> v] -> Mark = false;                 Pre[j -> v] = j;                 j -> Mark = true;                 Q.push(ES(j -> v, d[j -> v]));             }         }     } }   int DFS_1(int x, int Dep, int Fa) {     Depth[x] = Dep; Father[x] = Fa;     Size[x] = 1;     int SonSize, MaxSonSize;     SonSize = MaxSonSize = 0;     for (Edge *j = Point[x]; j; j = j -> Next) {         if (j -> v == Fa || j -> Mark == false) continue;         SonSize = DFS_1(j -> v, Dep + 1, x);         if (SonSize > MaxSonSize) {             MaxSonSize = SonSize;             Son[x] = j -> v;         }         Size[x] += SonSize;     }     return Size[x]; }   void DFS_2(int x) {     if (x == 0) return;     if (x == Son[Father[x]]) Top[x] = Top[Father[x]];     else Top[x] = x;     Pos[x] = ++Index;     DFS_2(Son[x]);     for (Edge *j = Point[x]; j; j = j -> Next) {         if (j -> v == Father[x] || j -> v == Son[x] || j -> Mark == false) continue;         DFS_2(j -> v);     } }   void Build_Tree(int x, int s, int t) {     D[x] = INF;     if (s == t) return;     int m = (s + t) >> 1;     Build_Tree(x << 1, s, m);     Build_Tree(x << 1 | 1, m + 1, t); }   void Init_LCA() {     for (int i = 1; i <= n; ++i) Jump[i][0] = Father[i];     for (int j = 1; j <= MaxLog; ++j) {         for (int i = 1; i <= n; ++i) {             if (Depth[i] < (1 << j)) continue;             Jump[i][j] = Jump[Jump[i][j - 1]][j- 1];         }     } }   int LCA(int x, int y) {     int Dif;     if (Depth[x] < Depth[y]) swap(x, y);     Dif = Depth[x] - Depth[y];     if (Dif) {         for (int i = 0; i <= MaxLog; ++i) {             if (Dif & (1 << i)) x = Jump[x][i];         }     }     if (x == y) return x;     for (int i = MaxLog; i >= 0; --i) {         if (Jump[x][i] != Jump[y][i]) {             x = Jump[x][i];             y = Jump[y][i];         }     }     return Father[x]; }   inline int gmin(int a, int b) {return a < b ? a : b;}   void Paint(int x, int Num) {     if (Num >= D[x]) return;     D[x] = Num; }   void PushDown(int x) {     if (D[x] == INF) return;     Paint(x << 1, D[x]);     Paint(x << 1 | 1, D[x]);     D[x] = INF; }   void Change(int x, int s, int t, int l, int r, int Num) {     if (l <= s && r >= t) {         Paint(x, Num);         return;     }     PushDown(x);     int m = (s + t) >> 1;     if (l <= m) Change(x << 1, s, m, l, r, Num);     if (r >= m + 1) Change(x << 1 | 1, m + 1, t, l, r, Num); }   void EChange(int x, int y, int z) {     int fx, fy;     fx = Top[x]; fy = Top[y];     while (fx != fy) {         Change(1, 1, n, Pos[fx], Pos[x], z);         x = Father[fx];         fx = Top[x];     }     if (x != y) Change(1, 1, n, Pos[y] + 1, Pos[x], z); }   int Get(int x, int s, int t, int p) {     if (s == t) return D[x];     PushDown(x);     int m = (s + t) >> 1;     int ret;     if (p <= m) ret = Get(x << 1, s, m, p);     else ret = Get(x << 1 | 1, m + 1, t, p);     return ret; }   int main() {     scanf("%d%d", &n, &m);     int a, b, c;     for (int i = 1; i <= m; ++i) {         scanf("%d%d%d", &a, &b, &c);         AddEdge(a, b, c);         AddEdge(b, a, c);       }     Dijkstra();     DFS_1(1, 0, 0);     Index = 0;     DFS_2(1);     Build_Tree(1, 1, n);     Init_LCA();     int t;     for (Edge *j = E + 1; ; ++j) {         if (j -> Mark) continue;         t = LCA(j -> u, j -> v);         EChange(j -> v, t, d[j -> u] + j -> w + d[j -> v]);         if (j == P) break;     }     int Temp;     for (int i = 2; i <= n; ++i) {         Temp = Get(1, 1, n, Pos[i]);         if (Temp < INF) printf("%d\n", Temp - d[i]);         else printf("-1\n");     }     return 0; }