2021四川省省赛F. Direction Setting 拆边+费用流

Posted 2021-08-12 kaka0010

原题链接：https://codeforces.ml/gym/103117/problem/F

目录

• 题意
• 分析
• Code

题意

给你n个图m条边，你可以任意选择边的方向，使得
$$\sum _i^{n}max(0,di-ai)最小[di是点的入度]$$

分析

300的数据范围基本可以确定解法就是网络流，然后考虑怎么建边。当时做的时候把点拆成两个点，然后怎么选择边就不知道该怎么处理了。其实最直接有效的方法就是拆边，因为题目有二选一的性质，因此用边作为虚点，然后分别向两个顶点去连边，这样可以达成二选一的操作（思维上还是差点 ）。

下面还剩下ai没有用到，从式子中分析，对于每个点来说ai相当于免费的流量，超过ai的部分需额外加费用，那么我们直接从顶点拉两条线出来到汇点，一条是ai的流向0费用，一条是无限流量1费用。

其实思维难度上还远未到金牌题的难度，做到一半就放弃了还是不应该的。

Code

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef pair<int, int> PII;
const ll INF = 1e9;
const int N = 600 + 10;
const int M = 3e6 + 10;
const int MOD = 1e9 + 7;
int st, ed;
struct node {
    int maxflow, mincost, cnt;
    int vis[N], dis[N], pre[N], last[N], h[N], flow[N];
    struct edge {
        int to, next, cap, cos;
    } e[M << 1];

    void add(int u, int v, int cap, int cos) {
        e[cnt].to = v;
        e[cnt].cap = cap;
        e[cnt].cos = cos;
        e[cnt].next = h[u];
        h[u] = cnt++;

        e[cnt].to = u;
        e[cnt].cap = 0;
        e[cnt].cos = -cos;
        e[cnt].next = h[v];
        h[v] = cnt++;
    }

    void init() {
        memset(h, -1, sizeof h);
        cnt = 0;
        mincost = maxflow = 0;
    }
    bool spfa() {
        queue<int> q;
        for (int i = 0; i <= ed; i++) dis[i] = INF, vis[i] = 0;
        vis[st] = 1, dis[st] = 0, flow[st] = INF;
        q.push(st);
        while (q.size()) {
            int u = q.front();
            q.pop(); vis[u] = 0;
            for (int i = h[u]; ~i; i = e[i].next) {
                int v = e[i].to;
                if (e[i].cap && dis[v] > dis[u] + e[i].cos) {
                    dis[v] = e[i].cos + dis[u];
                    flow[v] = min(flow[u], e[i].cap);
                    pre[v] = u;
                    last[v] = i;
                    if (!vis[v]) {
                        vis[v] = 1;
                        q.push(v);
                    }
                }
            }
        }
        if (dis[ed] != INF) return true;
        else return false;
    }

    void MCMF() {
        while (spfa()) {
            int now = ed;
            maxflow += flow[ed];
            mincost += flow[ed] * dis[ed];

            while (st != now) {
                e[last[now]].cap -= flow[ed];
                e[last[now] ^ 1].cap += flow[ed];
                now = pre[now];
            }
        }
    }
}mcmf;
int a[N], aa[N], bb[N];
void solve() {
    int T; cin >> T; while (T--) {
        int n, m; cin >> n >> m;
        mcmf.init();
        st = n + m + 1; ed = n + m + 2;
        for (int i = 1; i <= n; i++) cin >> a[i];
        for (int i = 1; i <= m; i++) {
            cin >> aa[i] >> bb[i];
            mcmf.add(st, i + n, 1, 0);
            mcmf.add(i+n, aa[i], 1, 0);
            mcmf.add(i+n, bb[i], 1, 0);
        }
        for (int i = 1; i <= n; i++) {
            mcmf.add(i, ed, a[i], 0);
            mcmf.add(i, ed, INF, 1);
        }
        mcmf.MCMF();
        cout << mcmf.mincost << endl;
        for (int i = n+1; i <= n+m; i++) {
            for (int j = mcmf.h[i]; ~j; j = mcmf.e[j].next) {
                int v = mcmf.e[j].to;
                if (v > n + m) continue;
                if (!mcmf.e[j].cap) {
                    if (v == bb[i-n]) cout << 0;
                    else cout << 1;
                    break;
                }
            }
        }
        cout << endl;
    }

}
signed main() {
    ios_base::sync_with_stdio(false);
    cin.tie(0);
    cout.tie(0);
#ifdef ACM_LOCAL
    freopen("input", "r", stdin);
    freopen("output", "w", stdout);
    signed test_index_for_debug = 1;
    char acm_local_for_debug = 0;
    do {
        if (acm_local_for_debug == '$') exit(0);
        if (test_index_for_debug > 20)
            throw runtime_error("Check the stdin!!!");
        auto start_clock_for_debug = clock();
        solve();
        auto end_clock_for_debug = clock();
        cout << "Test " << test_index_for_debug << " successful" << endl;
        cerr << "Test " << test_index_for_debug++ << " Run Time: "
             << double(end_clock_for_debug - start_clock_for_debug) / CLOCKS_PER_SEC << "s" << endl;
        cout << "--------------------------------------------------" << endl;
    } while (cin >> acm_local_for_debug && cin.putback(acm_local_for_debug));
#else
    solve();
#endif
    return 0;
}