[IndiaHacks 2016 - Online Edition (Div. 1 + Div. 2)] -D. Delivery Bears （二分+最大流）

題面：

題意：

給定一個含有\(\mathit n\)個節點\(\mathit m\)個有向邊的圖和\(\mathit x\)個工作小熊。保證有一個\(1->n\)的路徑。

現在讓你選擇一個最大的運輸總值，滿足當這\(\mathit x\)個小熊均分任務後選擇一些路徑從\(1->n\)，使其可以滿足

每一個有向邊的容量不小於走過該邊的小熊的任務總量。

思路：

顯然可以在區間\([1e(-6),1e6]\)這個區間去二分出每一個小熊最大的運輸任務。

對於check當前二分的數值\(mid\)，我們需要用到網路流中的最大流演算法，

在原圖的基礎上將邊權\(c_i\)改為\(c_i/mid\)向下取整的結果，跑最大流演算法，得出\(1->n\)的最大流量\(res\)，

判斷\(res\)與\(\mathit x\)的關係即可知道轉移的方向。

我才用ISAP演算法求最大流：該演算法的時間複雜度上限：\(O(E*V^2)\)，理想情況：\(O(sqrt(E)*V^2)\)

於是本題的ac程式碼時間複雜度為\(O(m*n^2*log_2x)\)

程式碼：

#include<bits/stdc++.h>
using namespace std;
#define N 1000
#define INF 1e15
typedef long long ll;
struct Edge {
    int from, to;
    ll cap, flow;
};
struct ISAP {
    int n, m, s, t;
    std::vector<Edge> edges;
    std::vector<int> G[N];
    bool vis[N];
    int d[N], cur[N];
    int p[N], num[N];
    void addedge(int from, int to, ll cap)
    {
        edges.push_back((Edge) {from, to, cap, 0});
        edges.push_back((Edge) {to, from, 0, cap});
        int m = edges.size();
        G[from].push_back(m - 2);
        G[to].push_back(m - 1);
    }
    void init()
    {
        memset(d, 0, sizeof(d));
        edges.clear();
        for (int i = 0; i <= n; ++i) {
            G[i].clear();
        }
    }
    ll Augument()
    {
        ll x = t, a = INF;
        while (x != s) {
            Edge &e = edges[p[x]];
            a = min(a, e.cap - e.flow);
            x = edges[p[x]].from;
        }
        x = t;
        while (x != s) {
            edges[p[x]].flow += a;
            edges[p[x] ^ 1].flow -= a;
            x = edges[p[x]].from;
        }
        return a;
    }
    void bfs()
    {
        memset(vis, 0, sizeof(vis));
        queue<int> q;
        q.push(t);
        d[t] = 0;
        vis[t] = 1;
        while (!q.empty()) {
            int x = q.front();
            q.pop();
            int len = G[x].size();
            for (int i = 0; i < len; ++i) {
                Edge &e = edges[G[x][i]];
                if (!vis[e.from] && e.cap > e.flow) {
                    vis[e.from] = 1;
                    d[e.from] = d[x] + 1;
                    q.push(e.from);
                }
            }
        }
    }
    ll Maxflow(int s, int t)
    {
        this->s = s;
        this->t = t;
        ll flow = 0;
        bfs();
        memset(num, 0, sizeof(num));
        for (int i = 0; i < n; ++i) {
            num[d[i]]++;
        }
        int x = s;
        memset(cur, 0, sizeof(cur));
        while (d[s] < n) {
            if (x == t) {
                flow += Augument();
                x = s;
            }
            int ok = 0;
            for (int i = cur[x]; i < G[x].size(); ++i) {
                Edge &e = edges[G[x][i]];
                if (e.cap > e.flow && d[x] == d[e.to] + 1) {
                    ok = 1;
                    p[e.to] = G[x][i];
                    cur[x] = i;
                    x = e.to;
                    break;
                }
            }
            if (!ok) {
                int m = n - 1;
                for (int i = 0; i < G[x].size(); ++i) {
                    Edge &e = edges[G[x][i]];
                    if (e.cap > e.flow) {
                        m = min(m, d[e.to]);
                    }
                }
                if (--num[d[x]] == 0) {
                    break;
                }
                num[d[x] = m + 1]++;
                cur[x] = 0;
                if (x != s) {
                    x = edges[p[x]].from;
                }
            }
        }
        return flow;
    }
} gao;

int n;
int m;
int x;
int a[N], b[N], c[N];

bool check(double mid)
{
    int S = 1;
    int T = n;
    gao.n = T + 1;

    for (int i = 1; i <= m; ++i) {
        ll num = floor(1.0 * c[i] / mid);
        gao.addedge(a[i], b[i], num);
    }
    int res = gao.Maxflow(S, T);
    gao.init();
    return res >= x;
}
int main()
{
    scanf("%d %d %d", &n, &m, &x);
    for (int i = 1; i <= m; ++i) {
        scanf("%d %d %d", &a[i], &b[i], &c[i]);
    }
    double l = 1e-6;
    double r = 1000000;
    double ans, mid;
    for(int rp=1;rp<=100;++rp)
    {
        mid = (l + r) * 0.5;
        if (check(mid)) {
            l = mid;
            ans = mid;
        } else {
            r = mid;
        }
    }
    ans*=x;
    printf("%.6f\n", ans );
    return 0;
}