poj 2112 Optimal Milking

Posted 2020-09-29 ZefengYao

Optimal Milking

Time Limit: 2000MS		Memory Limit: 30000K
Total Submissions: 17933		Accepted: 6415
Case Time Limit: 1000MS

Description

FJ has moved his K (1 <= K <= 30) milking machines out into the cow pastures among the C (1 <= C <= 200) cows. A set of paths of various lengths runs among the cows and the milking machines. The milking machine locations are named by ID numbers 1..K; the cow locations are named by ID numbers K+1..K+C. 

Each milking point can "process" at most M (1 <= M <= 15) cows each day. 

Write a program to find an assignment for each cow to some milking machine so that the distance the furthest-walking cow travels is minimized (and, of course, the milking machines are not overutilized). At least one legal assignment is possible for all input data sets. Cows can traverse several paths on the way to their milking machine. 

Input

* Line 1: A single line with three space-separated integers: K, C, and M. 

* Lines 2.. ...: Each of these K+C lines of K+C space-separated integers describes the distances between pairs of various entities. The input forms a symmetric matrix. Line 2 tells the distances from milking machine 1 to each of the other entities; line 3 tells the distances from machine 2 to each of the other entities, and so on. Distances of entities directly connected by a path are positive integers no larger than 200. Entities not directly connected by a path have a distance of 0. The distance from an entity to itself (i.e., all numbers on the diagonal) is also given as 0. To keep the input lines of reasonable length, when K+C > 15, a row is broken into successive lines of 15 numbers and a potentially shorter line to finish up a row. Each new row begins on its own line. 

Output

A single line with a single integer that is the minimum possible total distance for the furthest walking cow. 

Sample Input

2 3 2
0 3 2 1 1
3 0 3 2 0
2 3 0 1 0
1 2 1 0 2
1 0 0 2 0

Sample Output

2

翻译：一共K台挤奶机，每台最多可以给M头牛提供服务，现在一共有C头牛，在使得C头牛全部能产奶的情况下最小化牛的最大路程。
思路：根据题设给定的直接距离的邻接矩阵，先通过floyd预处理任意两个对象之间的最短距离。之后对最大路程进行二分处理，对于每一个路程，判断最大流是否为牛的个数C.
图的建立：设源点s，汇点t,s向每头牛连一条流量为1的边，每台机器向汇点t连一条流量为M的边，若每头牛ci能在最大路程内走到机器kj,则牛向机器连一条流量为1的边。
AC代码：

#define _CRT_SECURE_NO_DEPRECATE
#include<iostream>
#include<algorithm>
#include<queue>
#include<set>
#include<vector>
#include<cstring>
#include<string>
using namespace std;
#define INF 0x3f3f3f3f
const int K_MAX = 30+2, C_MAX = 200+2, V_MAX = K_MAX+C_MAX + 2;
typedef long long ll;
int V;
struct edge {
    int to, cap, rev;
    edge(int to, int cap, int rev) :to(to), cap(cap), rev(rev) {}
};
vector<edge>G[V_MAX];
int level[V_MAX];
int iter[V_MAX];

void add_edge(int from, int to, int cap) {
    G[from].push_back(edge(to, cap, G[to].size()));
    G[to].push_back(edge(from, 0, G[from].size() - 1));
}

void bfs(int s) {
    memset(level, -1, sizeof(level));
    queue<int>que;
    level[s] = 0;
    que.push(s);
    while (!que.empty()) {
        int v = que.front(); que.pop();
        for (int i = 0; i < G[v].size(); i++) {
            edge&e = G[v][i];
            if (e.cap > 0 && level[e.to] < 0) {
                level[e.to] = level[v] + 1;
                que.push(e.to);
            }
        }
    }
}

int dfs(int v, int t, int f) {
    if (v == t)return f;
    for (int &i = iter[v]; i < G[v].size(); i++) {
        edge&e = G[v][i];
        if (e.cap > 0 && level[v] < level[e.to]) {
            int d = dfs(e.to, t, min(f, e.cap));
            if (d > 0) {
                e.cap -= d;
                G[e.to][e.rev].cap += d;
                return d;
            }
        }
    }
    return 0;
}

int max_flow(int s, int t) {
    int flow = 0;
    for (;;) {
        bfs(s);
        if (level[t] < 0)return flow;
        memset(iter, 0, sizeof(iter));
        int f;
        while ((f = dfs(s, t, INT_MAX))>0) {
            flow += f;
        }
    }
}

int k, c, m,s,t;
int d[K_MAX+C_MAX][K_MAX+C_MAX];

void floyd() {
    for (int K = 0; K < k + c; K++)
        for (int i = 0; i < k + c; i++)
            for (int j = 0; j < k + c; j++)
                d[i][j] = min(d[i][j],d[i][K]+d[K][j]);
}

bool C(int limit) {
    for (int i = 0; i < V;i++) {
        G[i].clear();
    }
    for (int i = 0; i < c;i++) {
        add_edge(s,i,1);
    }
    for (int i = 0; i < k;i++) {
        add_edge(c+i,t,m);
    }
    for (int i = 0; i < c;i++) {//牛与机器连边,i:牛
        for (int j = 0; j < k;j++) {//j:机器
            if (d[j][k+i] <= limit)
                add_edge(i, c + j, 1);
        }
    }

    return max_flow(s,t)==c;//如果最大流等于牛的个数

}

int main() {
    while (scanf("%d%d%d", &k, &c, &m) != EOF) {
        for (int i = 0; i < k + c; i++)
            for (int j = 0; j < k + c; j++) {
                scanf("%d", &d[i][j]);
                if (!d[i][j])d[i][j] = INF;//!!Entities not directly connected by a path have a distance of 0
            }
        floyd();
        //0~c-1:牛
        //c~c+k-1:挤奶机
        s = k + c, t = s+1,V=t+1;

        int lb = 0, ub = 200 * (k + c);
        while (ub - lb > 1) {
            int mid = (lb+ub) >> 1;
            if (C(mid))ub = mid;
            else lb = mid;
        }
        printf("%d\n",ub);
    }
    return 0;
}