POJ 1149 PIGS

PIGS

Time Limit: 1000MS		Memory Limit: 10000K
Total Submissions: 20579		Accepted: 9387

Description

Mirko works on a pig farm that consists of M locked pig-houses and Mirko can't unlock any pighouse because he doesn't have the keys. Customers come to the farm one after another. Each of them has keys to some pig-houses and wants to buy a certain number of pigs. 
All data concerning customers planning to visit the farm on that particular day are available to Mirko early in the morning so that he can make a sales-plan in order to maximize the number of pigs sold. 
More precisely, the procedure is as following: the customer arrives, opens all pig-houses to which he has the key, Mirko sells a certain number of pigs from all the unlocked pig-houses to him, and, if Mirko wants, he can redistribute the remaining pigs across the unlocked pig-houses. 
An unlimited number of pigs can be placed in every pig-house. 
Write a program that will find the maximum number of pigs that he can sell on that day.

Input

The first line of input contains two integers M and N, 1 <= M <= 1000, 1 <= N <= 100, number of pighouses and number of customers. Pig houses are numbered from 1 to M and customers are numbered from 1 to N. 
The next line contains M integeres, for each pig-house initial number of pigs. The number of pigs in each pig-house is greater or equal to 0 and less or equal to 1000. 
The next N lines contains records about the customers in the following form ( record about the i-th customer is written in the (i+2)-th line): 
A K1 K2 ... KA B It means that this customer has key to the pig-houses marked with the numbers K1, K2, ..., KA (sorted nondecreasingly ) and that he wants to buy B pigs. Numbers A and B can be equal to 0.

Output

The first and only line of the output should contain the number of sold pigs.

Sample Input

3 3
3 1 10
2 1 2 2
2 1 3 3
1 2 6

Sample Output

7

Source

Croatia OI 2002 Final Exam - First day

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网络流——流量传递类型。

四月份的时候曾经写过一遍，但是已然忘光了，所以只得重新来过。

错误的建图方法

看到题之后有个简单的建图想法，就是设立M*N个猪圈点，以及N个商人点，如下建图：

源点向第一排M个猪圈点连容量为初始猪数目的边，代表一开始猪圈中最多存在的猪。

每排M个猪圈点中，A个能被打开的猪圈点向商人点连容量为无穷的边，代表可以被打开。

每排的商人点向汇点连容量为商人购买数目B的边，表示这个人的最大购买限度。

每排的商人点向下一排的A个点（这A个点还是本排商人能打开的猪圈）连无穷边，代表猪可以自由转移。

当然，因为猪也可以待在猪圈里不动，所以每排的各个猪圈向下一排的该猪圈连无穷边。

这样跑最大流，显然可以得到最优解，但是点的数量是O(N*M)的，而边的数量是……（懒得想了），果不其然地TLE了。

#include <cstdio>
#include <cstring>

inline int get_c(void)
{
    static const int siz = ;

    static char buf[siz];
    static char *head = buf + siz;
    static char *tail = buf + siz;

    if (head == tail)
        fread(head = buf, , siz, stdin);

    return *head++;
}

inline int get_i(void)
{
    register int ret = ;
    register int neg = false;
    register int bit = get_c();

    for (; bit < ; bit = get_c())
        if (bit == '-')neg ^= true;

    for (; bit > ; bit = get_c())
        ret = ret *  + bit - ;

    return neg ? -ret : ret;
}

template <class T>
inline T min(T a, T b)
{
    return a < b ? a : b;
}

const int inf = 2e9;
const int maxn = ;

int n, m;
int s, t;
int edges;
int hd[maxn];
int to[maxn];
int nt[maxn];
int fl[maxn];

inline void add(int u, int v, int f)
{    //printf("add %d %d %d\n", u, v, f);
    nt[edges] = hd[u]; to[edges] = v; fl[edges] = f; hd[u] = edges++;
    nt[edges] = hd[v]; to[edges] = u; fl[edges] = ; hd[v] = edges++;
}

int dep[maxn];

inline bool bfs(void)
{
    static int que[maxn];
    static int head, tail;

    memset(dep, , sizeof(dep));
    head = , tail = ;
    que[tail++] = s;
    dep[s] = ;

    while (head != tail)
    {
        int u = que[head++], v;
        for (int i = hd[u]; ~i; i = nt[i])
            if (!dep[v = to[i]] && fl[i])
            {
                dep[v] = dep[u] + ;
                que[tail++] = v;
            }
    }

    return dep[t] != ;
}

int dfs(int u, int f)
{
    if (u == t || !f)
        return f;

    int used = , flow, v;

    for (int i = hd[u]; ~i; i = nt[i])
        if (dep[v = to[i]] == dep[u] +  && fl[i])
        {
            flow = dfs(v, min(f - used, fl[i]));

            used += flow;
            fl[i] -= flow;
            fl[i^] += flow;

            if (used == f)
                return f;
        }

    if (!used)
        dep[u] = ;

    return used;
}

inline int maxFlow(void)
{
    int maxFlow = , newFlow;

    while (bfs())
        while (newFlow = dfs(s, inf))
            maxFlow += newFlow;

    return maxFlow;
}

signed main(void)
{
    n = get_i();
    m = get_i();

    s = , t = (n + ) * m + ;

    memset(hd, -, sizeof(hd));

    for (int i = ; i <= n; ++i)
    {
        int pigs = get_i();
        add(s, i, pigs);
    }

    for (int i = ; i < m; ++i)
        for (int j = ; j <= n; ++j)
            add((n + )*(i - ) + j, (n + )*i + j, inf);

    for (int i = ; i <= m; ++i)
    {
        int k = get_i();
        for (int j = ; j <= k; ++j)
        {
            int p = get_i();
            add((n + )*(i - ) + p, (n + )*i, inf);
            add((n + )*i, (n + )*i + p, inf);
        }
        add((n + )*i, t, get_i());
    }

    printf("%d\n", maxFlow());
}

正确的建图方法

设置一个源点，向M个猪圈点连初始数目的边，代表一开始的猪的数目。

然后想办法简化猪圈之间的转移，着重商人能得到的猪的来源，考虑直接在商人之间进行转移。

易知，如果商人X和商人Y，有X比Y先来，且X和Y有公共的猪圈的钥匙，那么Y能享受X能到达的所有猪圈。

用last_i表示上一个可以打开i猪圈的点，初始last_i=i猪圈初始点。

一个商人，如果有猪圈k的钥匙，那么last_k的所有流量都可以向这个商人转移，所有从last_k向商人连边，同时把last_k设为该商人。

商人点向汇点连最大购买数量的边。

跑最大流即可，点数O(N+M)，边数O(N+M)，小菜一碟了。

 #include <cstdio>
 #include <cstring>

 inline int get_c(void)
 {
     static const int siz = ;

     static char buf[siz];
     static char *head = buf + siz;
     static char *tail = buf + siz;

     if (head == tail)
         fread(head = buf, , siz, stdin);

     return *head++;
 }

 inline int get_i(void)
 {
     register int ret = ;
     register int neg = false;
     register int bit = get_c();

     for (; bit < ; bit = get_c())
         if (bit == '-')neg ^= true;

     for (; bit > ; bit = get_c())
         ret = ret *  + bit - ;

     return neg ? -ret : ret;
 }

 template <class T>
 inline T min(T a, T b)
 {
     return a < b ? a : b;
 }

 const int inf = 2e9;
 const int maxn = ;

 int n, m;
 int s, t;
 int edges;
 int hd[maxn];
 int to[maxn];
 int nt[maxn];
 int fl[maxn];

 inline void add(int u, int v, int f)
 {    //printf("add %d %d %d\n", u, v, f);
     nt[edges] = hd[u]; to[edges] = v; fl[edges] = f; hd[u] = edges++;
     nt[edges] = hd[v]; to[edges] = u; fl[edges] = ; hd[v] = edges++;
 }

 int dep[maxn];

 inline bool bfs(void)
 {
     static int que[maxn];
     static int head, tail;

     memset(dep, , sizeof(dep));
     head = , tail = ;
     que[tail++] = s;
     dep[s] = ;

     while (head != tail)
     {
         int u = que[head++], v;
         for (int i = hd[u]; ~i; i = nt[i])
             if (!dep[v = to[i]] && fl[i])
             {
                 dep[v] = dep[u] + ;
                 que[tail++] = v;
             }
     }

     return dep[t] != ;
 }

 int dfs(int u, int f)
 {
     if (u == t || !f)
         return f;

     int used = , flow, v;

     for (int i = hd[u]; ~i; i = nt[i])
         if (dep[v = to[i]] == dep[u] +  && fl[i])
         {
             flow = dfs(v, min(f - used, fl[i]));

             used += flow;
             fl[i] -= flow;
             fl[i^] += flow;

             if (used == f)
                 return f;
         }

     if (!used)
         dep[u] = ;

     return used;
 }

 inline int maxFlow(void)
 {
     int maxFlow = , newFlow;

     while (bfs())
         while (newFlow = dfs(s, inf))
             maxFlow += newFlow;

     return maxFlow;
 }

 int last[maxn];

 signed main(void)
 {
     n = get_i();
     m = get_i();

     s = , t = n + m + ;

     memset(hd, -, sizeof(hd));

     for (int i = ; i <= n; ++i)
         add(s, i, get_i()), last[i] = i;

     for (int i = ; i <= m; ++i)
     {
         for (int j = get_i(), k; j--; )
             k = get_i(), add(last[k], n + i, inf), last[k] = n + i;
         add(n + i, t, get_i());
     }

     printf("%d\n", maxFlow());
 }

@Author: YouSiki