[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

#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
", Temp - d[i]);
		else printf("-1
");
	}
	return 0;
}