[SDOI 2015] 星际战争

[题目链接]

         https://www.lydsy.com/JudgeOnline/problem.php?id=3993

[算法]

         首先发现问题具有单调性 ， 不妨二分答案mid 

         考虑网络流 :

         将源点向每个"激光武器”连一条流量为mid * Bi的边

         将每个“激光武器”向每个其可以攻击的“机器人”连一条流量为正无穷的边

         将每个“机器人”向汇点连一条流量为Ai的边

         判断是否满流即可

         时间复杂度 : O(dinic(N + M , M ^ 2) * logV)

[代码]

        为避免精度误差 ， 可以在整数域上进行二分 ， 最后以浮点数形式输出

        

#include<bits/stdc++.h>
using namespace std;
#define N 510
typedef long long ll;
typedef long double ld;
typedef unsigned long long ull;
const ll inf = 1e15;

struct edge
{
        int to;
        ll w;
        int nxt;
} e[N * N * 5];

int n , m , tot , S , T;
int dep[N] , head[N] , g[N][N];
ll a[N] , b[N];

template <typename T> inline void chkmax(T &x,T y) { x = max(x,y); }
template <typename T> inline void chkmin(T &x,T y) { x = min(x,y); }
template <typename T> inline void read(T &x)
{
    T f = 1; x = 0;
    char c = getchar();
    for (; !isdigit(c); c = getchar()) if (c == '-') f = -f;
    for (; isdigit(c); c = getchar()) x = (x << 3) + (x << 1) + c - '0';
    x *= f;
}
inline void addedge(int u , int v , ll w)
{
        ++tot;
        e[tot] = (edge){v , w , head[u]};
        head[u] = tot;
        ++tot;
        e[tot] = (edge){u , 0 , head[v]};
        head[v] = tot;
}
inline bool bfs()
{
        queue< int > q;
        for (int i = 1; i <= T; ++i)
                dep[i] = -1;
        q.push(S);
        dep[S] = 1;
        while (!q.empty())
        {
                int cur = q.front();
                q.pop();
                for (int i = head[cur]; i; i = e[i].nxt)
                {
                        int v = e[i].to;
                        ll w = e[i].w;
                        if (w > 0 && dep[v] == -1)
                        {
                                dep[v] = dep[cur] + 1;
                                q.push(v);
                                if (v == T) return true;
                        }
                }
        }
        return false;
}
inline ll dinic(int u , ll flow)
{
        ll k , rest = flow;
        if (u == T)
                return flow;
        for (int i = head[u]; i && rest; i = e[i].nxt)
        {
                int v = e[i].to;
                ll w = e[i].w;
                if (dep[v] == dep[u] + 1 && w)
                {
                        k = dinic(v , min(w , rest));
                        e[i].w -= k;
                        e[i ^ 1].w += k;
                        if (!k) dep[v] = 0;
                        rest -= k;
                }
        }
        return flow - rest;
}
inline bool check(ll mid)
{
        S = n + m + 1 , T = S + 1;
        for (int i = 1; i <= T; ++i) head[i] = 0;
        for (int i = 1; i <= tot; ++i) e[i].nxt = 0;
        tot = 1;
        for (int i = 1; i <= m; ++i) addedge(S , i , b[i] * mid);
        for (int i = 1; i <= m; ++i)
        {
                for (int j = 1; j <= n; ++j)
                {
                        if (g[i][j])
                                addedge(i , j + m , inf);
                }
        }
        ll sum = 0;
        for (int i = 1; i <= n; ++i) 
        {
                addedge(i + m , T , a[i]);
                sum += a[i];
        }
        ll res = 0;
        while (bfs())
        {
                while (double flow = dinic(S , inf)) res += flow;
        }
        return res == sum;
}

int main()
{
        
        scanf("%d%d" , &n , &m); 
        for (int i = 1; i <= n; ++i) 
        {
                scanf("%lld" , &a[i]);
                a[i] *= 1000;
        }
        for (int i = 1; i <= m; ++i) scanf("%lld" , &b[i]);
        for (int i = 1; i <= m; ++i)
        {
                for (int j = 1; j <= n; ++j)
                {
                        scanf("%d" , &g[i][j]);
                }
        }
        ll l = 0 , r = inf , ans = 0;
        while (l <= r)
        {
                int mid = (l + r) >> 1;
                if (check(mid))
                {
                        ans = mid;
                        r = mid - 1;
                } else l = mid + 1;
        }
        printf("%.6lf\n" , (double)(ans / 1000.0));
        
        return 0;
    
}