POJ 3422 Kaka's Matrix Travels 费用流

最新推荐文章于 2018-08-06 12:54:44 发布

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最新推荐文章于 2018-08-06 12:54:44 发布

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原文链接：http://www.cnblogs.com/ac-luna/p/3759966.html

这篇博客讨论了在给定矩阵中，玩家Kaka通过特定规则移动并累计最大收益的问题。通过拆点技巧和SPFA算法实现求解。

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Kaka's Matrix Travels

Time Limit: 1000MS		Memory Limit: 65536K
Total Submissions: 7465		Accepted: 3004

Description

On an N × N chessboard with a non-negative number in each grid, Kaka starts his matrix travels with SUM = 0. For each travel, Kaka moves one rook from the left-upper grid to the right-bottom one, taking care that the rook moves only to the right or down. Kaka adds the number to SUM in each grid the rook visited, and replaces it with zero. It is not difficult to know the maximum SUM Kaka can obtain for his first travel. Now Kaka is wondering what is the maximum SUM he can obtain after his Kth travel. Note the SUM is accumulative during the K travels.

Input

The first line contains two integers N and K (1 ≤ N ≤ 50, 0 ≤ K ≤ 10) described above. The following N lines represents the matrix. You can assume the numbers in the matrix are no more than 1000.

Output

The maximum SUM Kaka can obtain after his Kth travel.

Sample Input

Sample Output

Source

POJ Monthly--2007.10.06, Huang, Jinsong

题目大意：K取方格数

题目分析：拆点，每个点 i 拆成 i 、i'，因为每个格子的价值只能取一次，所以建边（i，i'，1，w），又因为每个格子能多次经过，所以建边（i，i'，oo，0）。对于其他的点，沿着路径建边即可。

代码如下：

#include <stdio.h>
#include <string.h>
#include <algorithm>
#define min(a, b) ((a) < (b) ? (a) : (b))
#define REP(i, n) for(int i = 1; i <= n; ++i)
#define MS0(X) memset(X,  0, sizeof X)
#define MS1(X) memset(X, -1, sizeof X)
using namespace std;
const int maxE = 3000000;
const int maxN = 5005;
const int maxM = 55;
const int oo = 0x3f3f3f3f;
struct Edge{
    int v, c, w, n;
};
Edge edge[maxE];
int adj[maxN], l;
int d[maxN], cur[maxN], Minflow;
int inq[maxN], Q[maxE], head, tail;
int cost, flow, s, t;
int n, m, nn, a[maxM][maxM];
void addedge(int u, int v, int c, int w){
    edge[l].v = v; edge[l].c = c; edge[l].w =  w; edge[l].n = adj[u]; adj[u] = l++;
    edge[l].v = u; edge[l].c = 0; edge[l].w = -w; edge[l].n = adj[v]; adj[v] = l++;
}
int SPFA(){
    memset(d, oo, sizeof d);
    memset(inq, 0, sizeof inq);
    head = tail = 0;
    d[s] = 0;
    Minflow = oo;
    cur[s] = -1;
    Q[tail++] = s;
    while(head != tail){
        int u =  Q[head++];
        inq[u] = 0;
        for(int i = adj[u]; ~i; i = edge[i].n){
            int v = edge[i].v;
            if(edge[i].c && d[v] > d[u] + edge[i].w){
                d[v] = d[u] + edge[i].w;
                cur[v] = i;
                Minflow = min(edge[i].c, Minflow);
                if(!inq[v]){
                    inq[v] = 1;
                    Q[tail++] = v;
                }
            }
        }
    }
    if(d[t] == oo) return 0;
    flow += Minflow;
    cost += Minflow * d[t];
    for(int i = cur[t]; ~i; i = cur[edge[i ^ 1].v]){
        edge[i].c -= Minflow;
        edge[i ^ 1].c += Minflow;
    }
    return 1;
}
int MCMF(){
    flow = cost = 0;
    while(SPFA());
    return cost;
}
void work(){
    REP(i, n) REP(j, n) scanf("%d", &a[i][j]);
    MS1(adj);
    l = 0;
    nn = n * n;
    s = 0;
    t = (nn << 1) + 1;
    addedge(s, 1, m, 0);
    addedge(nn << 1, t, m, 0);
    REP(i, n) REP(j, n){
        int ij = (i - 1) * n + j;
        addedge(ij, ij + nn, 1, -a[i][j]);
        addedge(ij, ij + nn, oo, 0);
        if(i < n) addedge(ij + nn, i * n + j, oo, 0);
        if(j < n) addedge(ij + nn, (i - 1) * n + j + 1, oo, 0);
    }
    printf("%d\n", -MCMF());
}
int main(){
    while(~scanf("%d%d", &n, &m)) work();
    return 0;
}