BZOJ1927: [Sdoi2010]星際競速(最小費用最大流 最小路徑覆蓋)

自為風月馬前卒發表於2018-12-06

題意

題目連結

Sol

看完題不難想到最小路徑覆蓋，但是帶權的咋做啊？qwqqq

首先冷靜思考一下：最小路徑覆蓋 = (n – ext{二分圖最大匹配數})

為什麼呢？首先最壞情況下是用(n)條路徑去覆蓋(就是(n)個點)，根據二分圖的性質，每個點只能有一個和他配對，這樣就保證了，每多出一個匹配，路徑數就會(-1)

擴充套件到有邊權的圖也是同理的，(i)表示二分圖左側的點,(i`)表示二分圖右側的點，對於兩點(u, v)，從(u)向(v`)連((1, w_i))的邊(前面是流量，後面是費用)

接下來從(S)向(i)連((1, 0))的邊，從(i`)向(T)連((1, 0))的邊，從(S)向(i`)連((1, A_i))的邊

跑最小費用最大流即可

#include<bits/stdc++.h>
#define chmin(x, y) (x = x < y ? x : y)
#define chmax(x, y) (x = x > y ? x : y)
using namespace std;
const int MAXN = 1e6 + 10, INF = 1e9 + 10;
inline int read() {
    char c = getchar(); int x = 0, f = 1;
    while(c < `0` || c > `9`) {if(c == `-`) f = -1; c = getchar();}
    while(c >= `0` && c <= `9`) x = x * 10 + c - `0`, c = getchar();
    return x * f;
}
int N, M, S, T, dis[MAXN], vis[MAXN], Pre[MAXN], ansflow, anscost, A[MAXN];
struct Edge {
    int u, v, f, w, nxt;
}E[MAXN];
int head[MAXN], num = 0;
inline void add_edge(int x, int y, int f, int w) {
    E[num] = (Edge) {x, y, f, w, head[x]}; head[x] = num++;    
}
inline void AddEdge(int x, int y, int f, int w) {
    add_edge(x, y, f, w); add_edge(y, x, 0, -w);
}
bool SPFA() {
    queue<int> q; q.push(S);
    memset(dis, 0x3f, sizeof(dis));
    memset(vis, 0, sizeof(vis));
    dis[S] = 0;
    while(!q.empty()) {
        int p = q.front(); q.pop(); vis[p] = 0;
        for(int i = head[p]; ~i; i = E[i].nxt) {
            int to = E[i].v;
            if(E[i].f && dis[to] > dis[p] + E[i].w) {
                dis[to] = dis[p] + E[i].w; Pre[to] = i;
                if(!vis[to]) vis[to] = 1, q.push(to);
            }
        }
    }
    return dis[T] <= INF;
}
void F() {
    int canflow = INF;
    for(int i = T; i != S; i = E[Pre[i]].u) chmin(canflow, E[Pre[i]].f);
    for(int i = T; i != S; i = E[Pre[i]].u) E[Pre[i]].f -= canflow, E[Pre[i] ^ 1].f += canflow;
    anscost += canflow * dis[T];
}
void MCMF() {
    while(SPFA()) F();
}
int main() {   
    memset(head, -1, sizeof(head));
    N = read(); M = read(); S = N * 2 + 2, T = N * 2 + 3;
    for(int i = 1; i <= N; i++) A[i] =read(), AddEdge(S, i, 1, 0), AddEdge(i + N, T, 1, 0), AddEdge(S, i + N, 1, A[i]);
    for(int i = 1; i <= M; i++) {
        int x = read(), y = read(), v = read();
        if(x > y) swap(x, y);
        AddEdge(x, y + N, 1, v);
    }
    MCMF();
    printf("%d
", anscost);
    return 0;
}

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