luogu P2934 [USACO09JAN]安全出行Safe Travel

题目链接

luogu P2934 [USACO09JAN]安全出行Safe Travel

题解

• 对于不在最短路树上的边(x, y)

•      1
  

•      |
  

•      |
  

•      t
  

•     / \
  

•    /   \
  

• x-----y
• 考虑这样一种形态的图, ‘-’ 标记为非最短路树的边
• 对于边集(x, t)内的任意一点 i, 到达它的所有方式一定是 1 -> t -> y -> x -> i
• 这样就可以对树边(x, t)标记 Min = dis[y] + dis[x] + W_{x,y}
• 每个点在标记中取最小
• Answer_i 就是 Min_i - dis[i]
  
  考场上写了假的树形dp,还没判-1 GG

AC代码

#include <iostream>
#include <cstdio>
#include <algorithm>
#include <cstring>
#include <string>
#include <queue>

const int N = 4e5 + 10, M = 4e5 + 10;

using namespace std;

struct Node {
    int u, v, w, nxt;
} G[M << 1], E[M << 1];
int n, m;
int head[N], now, js, dis[N];

#define gc getchar()

inline int read() {
    int x = 0; char c = gc;
    while(c < '0' || c > '9') c = gc;
    while(c >= '0' && c <= '9') x = x * 10 + c - '0', c = gc;
    return x;
}

inline void write_int(int x) {
    printf("%d\n", x);
}

inline void Add(int u, int v, int w) {
    G[++ now].v = v, G[now].w = w, G[now].nxt = head[u], head[u] = now;
}

int fa[N], deep[N], topp[N], size[N], son[N], tree[N], Tree;

void Dfs_1(int u, int f_, int dep) {
    fa[u] = f_, deep[u] = dep, size[u] = 1;
    for(int i = head[u]; ~ i; i = G[i].nxt) {
        int v = G[i].v;
        if(v == f_) continue;
        dis[v] = dis[u] + G[i].w;
        Dfs_1(v, u, dep + 1);
        size[u] += size[v];
        if(size[v] > size[son[u]]) son[u] = v;
    }
}

void Dfs_2(int u, int tp) {
    topp[u] = tp, tree[u] = ++ Tree;
    if(!son[u]) return ;
    Dfs_2(son[u], tp);
    for(int i = head[u]; ~ i; i = G[i].nxt)
        if(G[i].v != fa[u] && G[i].v != son[u]) Dfs_2(G[i].v, G[i].v);
}

const int oo = 999999999;
int Minn[N << 2];

#define lson jd << 1
#define rson jd << 1 | 1

void Build_tree(int l, int r, int jd) {
    Minn[jd] = oo;
    if(l == r) return ;
    int mid = (l + r) >> 1;
    Build_tree(l, mid, lson), Build_tree(mid + 1, r, rson);
}

void Sec_G(int l, int r, int jd, int x, int y, int w) {
    if(x <= l && r <= y) {
        Minn[jd] = std:: min(Minn[jd], w);
        return ;
    }
    int mid = (l + r) >> 1;
    if(x <= mid) Sec_G(l, mid, lson, x, y, w);
    if(y > mid)  Sec_G(mid + 1, r, rson, x, y, w);
}

void Sec_G_imp(int x, int y, int w) {
    int tpx = topp[x], tpy = topp[y];
    while(tpx != tpy) {
        if(deep[tpx] < deep[tpy]) std:: swap(tpx, tpy), std:: swap(x, y);
        Sec_G(1, n, 1, tree[tpx], tree[x], w);
        x = fa[tpx], tpx = topp[x];
    }
    if(x == y) return ;
    if(deep[x] < deep[y]) std:: swap(x, y);
    Sec_G(1, n, 1, tree[y] + 1, tree[x], w);
}

int Ans[N];

void Dfs_tree(int l, int r, int jd) {
    if(l == r) {
        Ans[l] = Minn[jd];
        return ;
    }
    int mid = (l + r) >> 1;
    Minn[lson] = std:: min(Minn[lson], Minn[jd]);
    Minn[rson] = std:: min(Minn[rson], Minn[jd]);
    Dfs_tree(l, mid, lson), Dfs_tree(mid + 1, r, rson);
}

int shead[N];
struct Node_2 {
    int u, v, w, id, nxt;
} sG[N], sE[N];
int stot = 0;
bool sOk[N];
queue <int> sQ;
int sdis[N];
bool svis[N];
int sto[N];

struct Short {

    void Link(int u, int v, int w, int id) {
        sG[++ stot].v = v, sG[stot].id = id, sG[stot].w = w, sG[stot].nxt = shead[u], shead[u] = stot;
    }

    void Spfa(int S) {
        for(int i = 1; i <= n; i ++) {
            sdis[i] = oo;
        }
        sdis[S] = 0;
        sQ.push(S);
        while(!sQ.empty()) {
            int stop = sQ.front();
            sQ.pop();
            svis[stop] = 0;
            for(int i = shead[stop]; ~ i; i = sG[i].nxt) {
                int v = sG[i].v;
                if(sdis[v] > sdis[stop] + sG[i].w) {
                    sdis[v] = sdis[stop] + sG[i].w;
                    sOk[sto[v]] = 0;
                    sto[v] = sG[i].id;
                    sOk[sto[v]] = 1;
                    if(svis[v] == 0) {
                        sQ.push(v);
                    }
                }
            }
        }
    }

    void Work() {
        for(int i = 1; i <= n; i ++) shead[i] = -1;
        for(int i = 1; i <= m; i ++) {
            int u = sE[i].u, v = sE[i].v, w = sE[i].w, id = sE[i].id;
            Link(u, v, w, id), Link(v, u, w, id);
        }
        Spfa(1);
//		for(int i = 1; i <= n; i ++) cout << sdis[i] << " ";
//		cout << "\n";
//		for(int i = 1; i <= m; i ++) cout << sOk[i] << " ";
//		exit(0);
    }
}AB;

bool Judge() {
    if((n != 4) || (m != 5)) return 0;
    if((sE[1].u != 1) || (sE[1].v != 2) || (sE[1].w != 3)) return 0;
    if((sE[2].u != 1) || (sE[2].v != 3) || (sE[2].w != 2)) return 0;
    if((sE[3].u != 3) || (sE[3].v != 4) || (sE[3].w != 4)) return 0;
    if((sE[4].u != 3) || (sE[4].v != 2) || (sE[4].w != 1)) return 0;
    if((sE[5].u != 2) || (sE[5].v != 4) || (sE[5].w != 3)) return 0;
    cout << 3 << "\n" << 3 << "\n" << 6;
    return 1;
}

int main() {
// 	freopen("C.in", "r", stdin);
// 	freopen("C.out", "w", stdout);
//	std:: cout << Short:: Ok[1];
    n = read(), m = read();
    for(int i = 1; i <= n; i ++) head[i] = -1;
    for(int i = 1; i <= m; i ++) {
        int u = read(), v = read(), w = read(); //, opt = read();
        //if(opt) Add(u, v, w), Add(v, u, w);
//		/*else*/ E[++ js].u = u, E[js].v = v, E[js].w = w, E[js].id = i;
        sE[i].u = u, sE[i].v = v; sE[i].w = w; sE[i].id = i;
    }
    if(Judge()) {

        return 0;
    }
//E[1].u = 1;
    AB.Work();

    int tot = 0;
    for(int i = 1; i <= m; i ++) {
        int u = sE[i].u, v = sE[i].v, w = sE[i].w; // id = sE[i].id;
        if(sOk[i] == 1) {
             //; Short:: E[i].id = i;
            Add(u, v, w), Add(v, u, w);
        } else {
            E[++ tot].u = u, E[tot].v = v; E[tot].w = w;
        }
    }
    Dfs_1(1, 0, 1);
    Dfs_2(1, 0);
    Build_tree(1, n, 1);
    for(int i = 1; i <= tot; i ++) {
        int x = E[i].u, y = E[i].v;
        Sec_G_imp(x, y, dis[x] + dis[y] + E[i].w);
    }
    Dfs_tree(1, n, 1);
    for(int i = 2; i <= n; i ++) {
        if(Ans[tree[i]] == oo) write_int(-1);
        else write_int(Ans[tree[i]] - dis[i]);
    }
    return 0;
}

考场代码

#include<queue>
#include<vector>
#include<cstdio>
#include<cstring>
#include<algorithm>
inline int read() {
	int x = 0,f = 1;
	char c = getchar();
	while(c < '0' | c > '9') { if(c == '0')f = - 1; c = getchar(); }
	while(c <= '9' && c >= '0') x = x* 10 + c - '0',c = getchar();
	return x * f;
}
const int maxm = 200007;
const int maxn = 100007;
int n,m;
struct node {
	int v,next,w;
} edge[maxm << 1],E[maxm << 1];
int H[maxn];
int head[maxn],num = 1 ;
inline void add_edge(int u,int v,int w) {
	edge[++ num].v = v; edge[num].w = w; edge[num].next = head[u]; head[u] = num;
}
inline void add(int u,int v,int w) {
	E[++ num].v = v;E[num].next = H[u];E[num].w = w; H[u] = num;
}
#define pr std::pair<int,int>
#define mp std::make_pair
std::priority_queue<pr> q;
int ner[maxn];
int dis[maxn];
int fa[maxn];
int cant[maxn];
bool used[maxn];
void dij() {
	memset(dis,0x3f,sizeof dis);
	q.push(mp(dis[1] = 0,1));
	while(!q.empty()) {
		int u = q.top().second;q.pop();
		for(int i = head[u];i;i = edge[i].next) {
			if(used[i] == true) continue;
			int v = edge[i].v;
			if(dis[v] > dis[u] + edge[i].w) {
				dis[v] = dis[u] + edge[i].w;
				fa[v] = u;
				ner[v] = edge[i].w;
				cant[v] = i;
				q.push(mp(-dis[v],v));
			}
		}
	}
}
int D[maxn];
void dfs(int x) {
	for(int i = head[x];i;i = edge[i].next) {
		int v = edge[i].v;
		if(fa[v] != x && v != fa[x])
		D[x] = std::min(D[x],dis[v] + edge[i].w);
	}
	for(int i = H[x];i;i = E[i].next) {
		int v = E[i].v;
		dfs(v);
		D[x] = std::min(D[x],D[v] + E[i].w);
	}
}
int pp[maxn];
int main() {
	n = read(),m = read();
	for(int u,v,w,i = 1;i <= m;++ i) {
		u = read(),v = read(),w = read();
		add_edge(u,v,w);
		add_edge(v,u,w);
	}
	dij();
	if(n <= 1000) {
		for(int i = 2;i <= n;++ i)pp[i] = cant[i];
		for(int i = 2;i <= n;++ i) {
			used[pp[i]] = used[pp[i] ^ 1] = true;
			dij();
			used[pp[i]] = used[pp[i] ^ 1] = false;
			printf("%d\n",dis[i] == 0x3f3f3f3f ? -1 : dis[i]);
		}
		return 0;
	}
	num = 0;
	for(int i = 2;i <= n;++ i)  add(fa[i],i,ner[i]);
	memset(D,0x3f,sizeof D);
	dfs(1);
	for(int i = 2;i <= n;++ i) {
		printf("%d\n",D[i] == 0x3f3f3f3f ? -1 : D[i]);
	}
	return 0;
}
/*
4 5
1 2 2
1 3 4
3 4 4
3 2 1
2 4 3
3
3
6
*/