2016 Multi-University Training Contest 9

 

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2016 Multi-University Training Contest 9

二分+最大权闭合图 Less Time, More profit（BH）

题意就是有n个工厂，m个商店 
每个工厂有建造时间ti，花费payi 
每个商店和k个工厂有关，如果这k个工厂都建造了，那么能获利proi 
问你求收益（∑pro−∑pay）≥L时，首先满足时间t最小，其次是收益p最大

首先二分时间的答案，然后看那些工厂能建造，然后工厂是花费，商店是收益，并且要与商店有关的工厂都建造了，才能获利，所以这是一个最大权闭合图的问题。

关于最大权闭合图：岐哥的博客

闭合图就是闭合图中的点的后继也在这个闭合图中。而闭合图是反映事物间必要条件的关系的，也就是说一个点是它后继的必要条件。也就是说要有这个点，它的后继点也要都存在。

最大权闭合图：

建图：构造一个源点S，汇点T。我们将S与所有权值为正的点连一条容量为其权值的边，将所有权值为负的点与T连一条容量为其权值的绝对值的边，原来的边将其容量定为正无穷。

求解：最大权闭合图=正的点权和-最小割=正的点权和-最大流。

时间复杂度为.

代码：

#include <bits/stdc++.h>

const int N = 200 + 5;
const int INF = 0x3f3f3f3f;
int pay[N], t[N];
int pro[N];
std::vector<int> need[N];
int n, m;
int L;

/*
   *最大流之Dinic算法：不停地构造层次图，然后用阻塞流增广。
   *时间复杂度为 O(n^2*m)。
   *如果容量为1，复杂度为 O(min(n^(2/3), m^(1/2)*m)。
   *对于二分图最大匹配这样的特殊图，复杂度为 O(n^(1/2)*m)
*/
struct Max_Flow {
    struct Edge {
        int from, to, cap, flow;
    };
    std::vector<Edge> edges;
    std::vector<int> G[2*N]; //保存每个结点的弧在edges里的序号
    int level[2*N], cur[2*N]; //结点在层次图中的等级。结点当前弧下标。
    int n, m, s, t;
    //初始化顶点个数n，边的个数m在加边时统计，s和t分别为源点和汇点
    void init(int n) {
        this->n = n;
        for (int i=0; i<=n; ++i) {
            G[i].clear ();
        }
        edges.clear ();
    }
    void add_edge(int from, int to, int cap) {
        edges.push_back ((Edge) {from, to, cap, 0});
        edges.push_back ((Edge) {to, from, 0, 0});
        m = edges.size ();
        G[from].push_back (m - 2);
        G[to].push_back (m - 1);
    }
    bool BFS() {
        std::fill (level, level+1+n, -1);
        std::queue<int> que;
        level[s] = 0; que.push (s);
        while (!que.empty ()) {
            int u = que.front (); que.pop ();
            for (int i=0; i<G[u].size (); ++i) {
                Edge &e = edges[G[u][i]];
                if (level[e.to] == -1 && e.cap > e.flow) {
                    level[e.to] = level[u] + 1;
                    que.push (e.to);
                }
            }
        }
        return level[t] != -1;
    }
    int DFS(int u, int a) {
        if (u == t || a == 0) {
            return a; //a表示当前为止所有弧的最小残量
        }
        int flow = 0, f;
        for (int &i=cur[u]; i<G[u].size (); ++i) {
            Edge &e = edges[G[u][i]];
            if (level[u] + 1 == level[e.to]
            && (f = DFS (e.to, std::min (a, e.cap - e.flow))) > 0) {
                e.flow += f;
                edges[G[u][i]^1].flow -= f;
                flow += f; a -= f;
                if (a == 0) {
                    break;
                }
            }
        }
        return flow;
    }
    int Dinic(int s, int t) {
        this->s = s; this->t = t;
        int flow = 0;
        while (BFS ())  {
            std::fill (cur, cur+1+n, 0);
            flow += DFS (s, INF);
        }
        return flow;
    }
}max_flow;

int check(int max_t) {
    int S = 0, T = n + m + 1;
    int sum = 0;
    max_flow.init (n+m+2);
    for (int i=1; i<=m; ++i) {
        max_flow.add_edge (S, i, pro[i]);
        bool flag = true;
        for (int j: need[i]) {
            if (t[j] > max_t) {
                flag = false;
                break;
            }
        }
        if (flag) {
            for (int j: need[i]) {
                max_flow.add_edge (i, m+j, INF);
            }
            sum += pro[i];
        }
    }
    for (int i=1; i<=n; ++i) {
        if (t[i] <= max_t)
            max_flow.add_edge (m+i, T, pay[i]);  //abs (-pay[i])
    }
    int min_cut = max_flow.Dinic (S, T);
    if (sum - min_cut >= L)
        return sum - min_cut;
    else
        return -1;
}

void solve(int cas) {
    printf ("Case #%d: ", cas);
    int best_t = INF, best_p = 0;
    int l = 1, r = 1000000000;
    while (l <= r) {
        int mid = l + r >> 1;
        int res = check (mid);
        if (res != -1) {
            best_t = std::min (best_t, mid);
            best_p = res;
            r = mid - 1;
        } else
            l = mid + 1;
    }
    if (best_t < INF)
        printf ("%d %d
", best_t, best_p);
    else
        puts ("impossible");
}

int main() {
    int T;
    scanf ("%d", &T);
    for (int cas=1; cas<=T; ++cas) {
        scanf ("%d%d%d", &n, &m, &L);
        for (int i=1; i<=n; ++i) {
            scanf ("%d%d", pay+i, t+i);
        }
        for (int i=1; i<=m; ++i) {
            scanf ("%d", pro+i);
            need[i].clear ();
            int k;
            scanf ("%d", &k);
            while (k--) {
                int id;
                scanf ("%d", &id);
                need[i].push_back (id);
            }
        }
        solve (cas);
    }
    return 0;
}

还有贪心的做法，时间复杂度，解释。

代码：

#include <bits/stdc++.h>

const int N = 200 + 5;
const int INF = 0x3f3f3f3f;

struct Plant {
    int pay, t;
}p[N];
std::vector<int> id[N];

struct Shop {
    int pro;
    std::vector<int> need;
    int pay;
    int max_t;
    int done;
}s[N];

int n, m;
int L;

bool vis_p[N], vis_s[N];

int check(int max_t) {
    memset (vis_p, false, sizeof (vis_p));
    memset (vis_s, false, sizeof (vis_s));
    for (int i=0; i<m; ++i)
        s[i].done = 0;
    int ret = -INF, tmp = 0;
    for (; ;) {
        int max_val = -INF, k = -1;
        for (int i=0; i<m; ++i) {
            if (!vis_s[i] && s[i].max_t <= max_t) {
                if (max_val < s[i].pro - s[i].pay + s[i].done) {
                    max_val = s[i].pro - s[i].pay + s[i].done;
                    k = i;
                }
            }
        }
        if (max_val == -INF)
            break;
        vis_s[k] = true;  //max m times
        tmp += max_val;
        ret = std::max (ret, tmp);
        for (int j: s[k].need) {
            if (!vis_p[j]) {
                vis_p[j] = true;
                for (int l: id[j]) {
                    if (!vis_s[l]) {
                        s[l].done += p[j].pay;
                    }
                }
            }
        }
    }
    return ret >= L ? ret : -1;
}

void solve(int cas) {
    printf ("Case #%d: ", cas);
    int best_t = INF, best_p = 0;
    int l = 1, r = 1000000000;
    while (l <= r) {
        int mid = l + r >> 1;
        int res = check (mid);
        if (res != -1) {
            best_t = std::min (best_t, mid);
            best_p = res;
            r = mid - 1;
        } else
            l = mid + 1;
    }
    if (best_t < INF)
        printf ("%d %d
", best_t, best_p);
    else
        puts ("impossible");
}

int main() {
    int T;
    scanf ("%d", &T);
    for (int cas=1; cas<=T; ++cas) {
        scanf ("%d%d%d", &n, &m, &L);
        for (int i=0; i<n; ++i) {
            scanf ("%d%d", &p[i].pay, &p[i].t);
            id[i].clear ();
        }
        for (int i=0; i<m; ++i) {
            scanf ("%d", &s[i].pro);
            s[i].need.clear ();
            s[i].max_t = -1;
            s[i].pay = 0;
            int k;
            scanf ("%d", &k);
            while (k--) {
                int j;
                scanf ("%d", &j);
                j--;
                if (p[j].t > s[i].max_t)
                    s[i].max_t = p[j].t;
                s[i].pay += p[j].pay;
                s[i].need.push_back (j);
                id[j].push_back (i);
            }
        }
        solve (cas);
    }
    return 0;
}