[学习笔记]插头dp

基于连通性的状压dp

巧妙之处：插头已经可以表示内部所有状态了。

就是讨论麻烦一些。

简介

转移方法：
逐格转移，分类讨论

记录状态方法：
最小表示法（每次要重新编号，对于一类没用“回路路径”之类的题，可以胜任）

括号表示法（便于操作，但是一些题不能记录状态）

状态存储方法：

不能直接循环所有可能状态，因为状态不满太浪费

哈希+滚动数组

（clear时候，直接memset(hd)，cnt=0就是最快的！！！！）

然后具体题目分清楚转移情况讨论即可。

例题

尝试加入各种剪枝以减少状态量。

经典入门例题：HDU 1693

Eat the Trees

代码：

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const int N=;
int n,m;
int mp[N+][N+];
int T;
ll f[N+][N+][<<(N+)];
ll ans;
void wrk(){
    for(int i=;i<=n;i++){
        for(int j=;j<=m;j++){
            for(int s=;s<(<<m+);s++){
                if(f[i][j][s]==) continue;

                 if(j!=m){// not a end
                     //cout<<" here "<<i<<" "<<j<<" "<<s<<endl;
                    int le=(s>>j-)&;
                    int up=(s>>j)&;
                    if(mp[i][j]==){
                        if(le==&&up==){
                            f[i][j+][s]+=f[i][j][s];
                        }
                    }
                    else{
                        int now=s;
                        if(le==&&up==){
                            int s0=s^(<<j-);
                            s0|=(<<j);
                            f[i][j+][s0]+=f[i][j][s];
                            f[i][j+][s]+=f[i][j][s];
                        }
                        else if(le==&&up==){
                            int s0=s^(<<j-);
                            s0^=(<<j);
                            f[i][j+][s0]+=f[i][j][s];
                        }
                        else if(le==&&up==){
                            int s0=s^(<<j);
                            s0|=(<<j-);
                            f[i][j+][s0]+=f[i][j][s];
                            f[i][j+][s]+=f[i][j][s];
                        }
                        else{
                            int s0=s|(<<j);
                            s0|=(<<j-);
                            f[i][j+][s0]+=f[i][j][s];
                        }
                    }
                }
                  else{// j==m
                      int le=(s>>j-)&;
                    int up=(s>>j)&;
                    if(mp[i][j]==){
                        if(le==&&up==){
                            f[i+][][(s<<)]+=f[i][j][s];
                        }
                    }
                    else{
                        int now=s;
                        //cout<<" irhf "<<i<<" "<<j<<" : "<<s<<" "<<le<<" "<<up<<endl;
                        if(le==&&up==){

                            f[i+][][s<<]+=f[i][j][s];
                        }
                        else if(le==&&up==){
                            int s0=s^(<<j-);
                            s0^=(<<j);
                            f[i+][][s0<<]+=f[i][j][s];
                        }
                        else if(le==&&up==){
                            int s0=s^(<<j);
                            s0|=(<<j-);
                            f[i+][][s0<<]+=f[i][j][s];
                        }
                        else{
                            continue;
                        }
                    }
                }
            }
        }
    }
}
int main()
{
    scanf("%d",&T);
    for(int o=;o<=T;o++){
        scanf("%d%d",&n,&m);
        for(int i=;i<=n;i++){
            for(int j=;j<=m;j++){
                scanf("%d",&mp[i][j]);
            }
        }
        ans=;
        memset(f,,sizeof f);
        f[][][]=;
        wrk();
        /*for(int i=1;i<=n;i++)
         for(int j=1;j<=m;j++){
             cout<<i<<" and "<<j<<" : "<<endl;
             for(int s=0;s<=(1<<m+1);s++){
                 cout<<" "<<s<<" |: "<<f[i][j][s]<<endl;
            }
         }*/

        ans=f[n+][][];
        printf("Case %d: There are %lld ways to eat the trees.\n",o,ans);
    }
    return ;
}

Eat the Trees

【模板】插头dp

https://www.luogu.org/problemnew/show/P5056

直接memset就是最好用的了。。。非要暴力清空结果TLE了一夜一页

对于下面和右面有障碍的，就不要填出插头了。剪枝

// luogu-judger-enable-o2
// luogu-judger-enable-o2
#include<bits/stdc++.h>
#define il inline
#define reg register int
#define numb (ch^'0')
using namespace std;
typedef long long ll;
il void rd(int &x){
    char ch;bool fl=false;
    while(!isdigit(ch=getchar()))(ch=='-')&&(fl=true);
    for(x=numb;isdigit(ch=getchar());x=x*+numb);
    (fl==true)&&(x=-x);
}
namespace Miracle{
const int N=;
const int M=;
const int mo=<<;
int n,m;
char mp[N][N];
int bit[N];
int lasx,lasy;
int num(int s,int k){//s's kth pos
    return (s>>bit[k-])&;
}
void zip(int s){
    for(reg i=;i<=m+;++i){
        cout<<num(s,i)<<" ";
    }cout<<endl;
}
struct hah{
    int val[M],hd[mo],nxt[M];
    ll dp[M];
    int cnt;
    void upda(int s,ll v){
        int p=s%mo;
//        cout<<" upda ";zip(s);
//        cout<<"hash "<<s<<" v "<<v<<" p "<<p<<" cnt "<<cnt<<" hd "<<hd[p]<<endl;
        for(reg i=hd[p];i;i=nxt[i]){
//            cout<<"i "<<i<<" nxt "<<nxt[i]<<endl;
            if(val[i]==s){
//                cout<<" fin "<<i<<endl;
                dp[i]+=v;return;
            }
        }
//        cout<<"build new "<<endl;
        ++cnt;nxt[cnt]=hd[p];dp[cnt]=v;val[cnt]=s;
        hd[p]=cnt;
    }
    void clear(){
        memset(hd,,sizeof hd);cnt=;
    }
}f[];

int ad(int k,int c){
//    cout<<" k "<<k<<" "<<c<<" bit "<<bit[k-1]<<endl;
    return (<<(bit[k-]))*c;
}
int get(int s,int k){
    int lp=num(s,k);
    if(lp==){//after
        int cnt=;
        for(reg i=k+;i<=m+;++i){
            if(num(s,i)==) ++cnt;
            else if(num(s,i)==) --cnt;
            if(cnt<) return i;
        }
    }else{//before
        int cnt=;
        for(reg i=k-;i>=;--i){
            if(num(s,i)==) --cnt;
            else if(num(s,i)==) ++cnt;
            if(cnt<) return i;
        }
    }
    cout<<" bug "<<endl;
    return -;//bug
}

ll wrk(){
    int tmp=;
    f[tmp].upda(,);
    ll lim=(<<(*(m+)))-;
    for(reg i=;i<=n;++i){
//        cout<<" i "<<i<<endl;
        for(reg j=;j<=f[tmp].cnt;++j) {
            f[tmp].val[j]=(f[tmp].val[j]*)&lim;
//            cout<<" num "<<j<<" : "<<" dp "<<f[tmp].dp[j]<<" zip ";zip(f[tmp].val[j]);
        }
        for(reg j=;j<=m;++j){
//            cout<<" j "<<j<<endl;
            tmp^=;
            f[tmp].clear();
            int pr=tmp^;
//            cout<<" cnt "<<f[pr].cnt<<endl;
            for(reg k=;k<=f[pr].cnt;++k){
//                cout<<" k "<<k<<endl;
                int s=f[pr].val[k];
//                int u=num(s,j+1),l;//=num(s,j);
                ll v=f[pr].dp[k];
//                cout<<" v "<<v;cout<<" s ";zip(s);//endl;
                if(mp[i][j]=='*'){
                    if(num(s,j)==&&num(s,j+)==) f[tmp].upda(s,v);
                }else{
                    if(num(s,j)==&&num(s,j+)==){//new
                        if(j!=m)f[tmp].upda(s+ad(j,)+ad(j+,),v);
                    }
                    else if(num(s,j)==){//only up
//                        cout<<" only up"<<endl;
                        f[tmp].upda(s+ad(j,num(s,j+))+ad(j+,-num(s,j+)),v);
                        if(j!=m)f[tmp].upda(s,v);
                    }else if(num(s,j+)==){//only left
//                        cout<<" only left "<<endl;
                        if(j!=m)f[tmp].upda(s+ad(j,-num(s,j))+ad(j+,num(s,j)),v);
                        f[tmp].upda(s,v);
//                        cout<<" after upda "<<endl;
                    }else{//both
//                    cout<<" both "<<endl;
                        if(num(s,j)==&&num(s,j+)==){
                            int to=s;to+=ad(j,-num(s,j))+ad(j+,-num(s,j+));
                            to+=ad(get(s,j+),-);
                            f[tmp].upda(to,v);
                        }else if(num(s,j)==&&num(s,j+)==){
                            int to=s;to+=ad(j,-num(s,j))+ad(j+,-num(s,j+));
                            to+=ad(get(s,j),);
                            f[tmp].upda(to,v);
                        }else if(num(s,j)==&&num(s,j+)==){
                            int to=s;to+=ad(j,-num(s,j))+ad(j+,-num(s,j+));
                            f[tmp].upda(to,v);
                        }else{
                            if(i==lasx&&j==lasy){
                                int to=s;to+=ad(j,-num(s,j))+ad(j+,-num(s,j+));
                                f[tmp].upda(to,v);
                            }
                        }
                    }
                }
            }
        }
    }
    for(reg i=;i<=f[tmp].cnt;++i){
        if(f[tmp].val[i]==) return f[tmp].dp[i];
    }
    return ;
}
int main(){
    rd(n);rd(m);
    for(reg i=;i<=;++i) bit[i]=i<<;
    for(reg i=;i<=n;++i){
        scanf("%s",mp[i]+);
    }
    for(reg i=;i<=n;++i){
        for(reg j=;j<=m;++j){
            if(mp[i][j]=='.') lasx=i,lasy=j;
        }
    }
    cout<<wrk();
    return ;
}

}
signed main(){
    Miracle::main();
    return ;
}

/*
   Author: *Miracle*
   Date: 2019/3/31 21:44:47
*/