【网络流】 HDU 1569 方格取数(2)

最新推荐文章于 2019-10-06 08:15:38 发布

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本文详细介绍了如何通过构建二分图并使用最大匹配算法来解决最大点权独立集与最小点权覆盖集的问题。通过引入特殊的边权重和源汇点设置，有效地将原始问题转化为求解最大流问题，进而找到最优解。

定理：最大点权独立集的点权和=点SUM-最小点权覆盖集的点权和

最小点覆盖=二分图的最大匹配

黑白染色分成一个二分图

黑色与相连的白色建一条INF的边

源点与黑色建一条边（容量为点值）

汇点与白色建一条边（容量为点值）

注意建边

#include <stdio.h>
#include <string.h>
#include <stdlib.h>
#include <string>
#include <iostream>
#include <algorithm>
#include <sstream>
#include <cmath>
using namespace std;
#include <queue>
#include <stack>
#include <vector>
#include <deque>
#include <set>
#include <map>
#include <time.h>;
#define cler(arr, val)    memset(arr, val, sizeof(arr))
#define FOR(i,a,b)  for(int i=a;i<=b;i++)
#define IN   freopen ("in.txt" , "r" , stdin);
#define OUT  freopen ("out.txt" , "w" , stdout);
typedef long long  LL;
const int MAXN = 2503;
const int MAXM = 15006;
const int INF = 0x3f3f3f3f;
const int mod = 10000007;
struct Edge
{
    int to,next,cap,flow;
} edge[MAXM]; //注意是MAXM
int tol;
int head[MAXN];
int gap[MAXN],dep[MAXN],cur[MAXN];
void init()
{
    tol = 0;
    memset(head,-1,sizeof (head));
}
void addedge (int u,int v,int w,int rw = 0)
{
    edge[tol].to = v;
    edge[tol].cap = w;
    edge[tol].flow = 0;
    edge[tol].next = head[u];
    head[u] = tol++;
    edge[tol].to = u;
    edge[tol].cap = rw;
    edge[tol].flow = 0;
    edge[tol].next = head[v];
    head[v] = tol++;
}
int Q[MAXN];
void BFS(int start,int end)
{
    memset(dep,-1,sizeof(dep));
    memset(gap,0,sizeof(gap));
    gap[0] = 1;
    int front = 0, rear = 0;
    dep[end] = 0;
    Q[rear++] = end;
    while(front != rear)
    {
        int u = Q[front++];
        for(int i = head[u]; i !=  -1; i = edge[i].next)
        {
            int v = edge[i]. to;
            if(dep[v] != -1)continue;
            Q[rear++] = v;
            dep[v] = dep[u] + 1;
            gap[dep[v]]++;
        }
    }
}
int S[MAXN];
int sap(int start,int end, int N)
{
    BFS(start,end);
    memcpy(cur,head,sizeof(head));
    int top = 0;
    int u = start;
    int ans = 0;
    int i;
    while(dep[start] < N)
    {
        if(u == end)
        {
            int Min = INF;
            int inser;
            for( i = 0; i < top; i++)
            {
                if(Min > edge[S[i]].cap - edge[S[i]].flow)
                {
                    Min = edge[S[i]].cap - edge[S[i]].flow;
                    inser = i;
                }
            }
            for( i = 0; i < top; i++)
            {
                edge[S[i]]. flow += Min;
                edge[S[i]^1].flow -= Min;
            }
            ans += Min;
            top = inser;
            u = edge[S[top]^1].to;
            continue;
        }
        bool flag =  false;
        int v;
        for( i = cur[u]; i != -1; i = edge[i]. next)
        {
            v = edge[i]. to;
            if(edge[i].cap - edge[i].flow && dep[v]+1 == dep[u])
            {
                flag =  true;
                cur[u] = i;
                break;
            }
        }
        if(flag)
        {
            S[top++] = cur[u];
            u = v;
            continue;
        }
        int Min = N;
        for( i = head[u]; i !=  -1; i = edge[i].next)
        {
            if(edge[i].cap - edge[i].flow && dep[edge[i].to] < Min)
            {
                Min = dep[edge[i].to];
                cur[u] = i;
            }
        }
        gap[dep[u]]--;
        if(!gap[dep[u]]) return ans;
        dep[u] = Min + 1;
        gap[dep[u]]++;
        if(u != start)u = edge[S[--top]^1].to;
    }
    return ans;
}
int n,m;
void addline(int x,int y)
{
    if(x+1<n)
        addedge(x*m+y,(x+1)*m+y,INF);
    if(x-1>=0)
        addedge(x*m+y,(x-1)*m+y,INF);
    if(y+1<m)
        addedge(x*m+y,x*m+y+1,INF);
    if(y-1>=0)
        addedge(x*m+y,x*m+y-1,INF);
}
int main()
{
    //IN;
    while(scanf("%d%d",&n,&m)!=EOF)
    {
        init();
        int sum=0;
        for(int i=0; i<n; i++)
        {
            for(int j=0; j<m; j++)
            {
                int mp;
                scanf("%d",&mp);
                sum+=mp;
                if((i+j)%2)
                {
                    addedge(n*m,i*m+j,mp);
                    addline(i,j);
                }
                else
                    addedge(i*m+j,n*m+1,mp);
            }
        }
        printf("%d\n",sum-sap(n*m,n*m+1,n*m+2));
    }
    return 0;
}