NOI2007生成树计数 状压DP+矩阵乘法

(http://www.elijahqi.win/2017/07/07/noi2007%E7%94%9F%E6%88%90%E6%A0%91%E8%AE%A1%E6%95%B0-%E7%8A%B6%E5%8E%8Bdp%E7%9F%A9%E9%98%B5%E4%B9%98%E6%B3%95/)

#include<cstdio>
#include<cstring>
#include<cmath>
#include<algorithm>
using namespace std;
int mat[3300][3300],k,n,b[3300];
long long ans;
bool get[3300];
void work(){
    //for (int i=1;i<=n;++i) printf("%d ",b[i]);printf("\n");
    long long t=0;
    for (int i=1;i<=n;++i)
        for (int j=i+1;j<=n;++j) if (b[i]>b[j]) ++t; 
    long long s;
    if ((t&1)==1) s=-1;else s=1;
    long long tmp=1;
    for (int i=1;i<=n;++i){
        tmp*=mat[i][b[i]];
    }
    ans+=tmp*s;
}
void find(int x){
    if (x==n+1){
        work();return;
    }
    for (int i=1;i<=n;++i){
        if (!get[i]&&mat[x][i]!=0){
            b[x]=i;get[i]=true;find(x+1);
            b[x]=0;get[i]=false;
        }
    }
}
int main(){
    //freopen("count.in","r",stdin);
    //freopen("count.out","w",stdout);
    scanf("%d%d",&k,&n);
    for (int i=1;i<=n;++i){
        for (int j=1;j<=n;++j){
            if (i==j) {mat[i][j]=i-max(1,i-k)+min(n,k+i)-i;continue;}//前面多少连过来+后面多少连过去 
            if (abs(i-j)<=k) {mat[i][j]=-1;continue;}
            mat[i][j]=0;
        }
    }
    n--;
    memset(get,false,sizeof(get));
    find(1);
    /*for (int i=1;i<=n;++i){
        for (int j=1;j<=n;++j){
            printf("%d ",mat[i][j]);
        }
        printf("\n");
    }*/
    printf("%lld ",ans);
    //printf("%d ",-5%4);
    return 0;
}

矩阵加速：ORZ纯手打代码没有参考别人题解 所以大概很长…

#include <cstdio>
#include <cstring>
#define N 65521
int k,b[60][10],num,numm[10],a[10],s[60],sh[1000000],link[10],father[10],map[60][60];
long long n;
bool flag[10];
int NUM[10]={0,1,1,3,16,125};
struct matrix{
    long long f[60][60],l,c;
}base1;
inline matrix multiply(matrix a,matrix b){
    matrix c;memset(c.f,0,sizeof(c.f));
    c.l=a.l;c.c=b.c;
    for (int i=1;i<=c.l;++i)
        for (int j=1;j<=c.c;++j)
            for (int z=1;z<=a.c;++z){
                c.f[i][j]+=a.f[i][z]*b.f[z][j]%N;
                c.f[i][j]%=N;
            }
    return c;
}
inline matrix pow(matrix a,long long b){
    matrix r=base1,base=a;
    while (b!=0){
        if (b&1) r=multiply(base,r);
        base=multiply(base,base);
        b>>=1;
    }
    return r;
}
void dfs(int x,int p){
    if (x>k){
        ++num;
        for (int i=1;i<=k;++i) b[num][i]=a[i];return;
    }
    for (int i=1;i<=k;++i){
        if (i<=p){
            a[x]=i;
            if (i==p) dfs(x+1,i+1);else dfs(x+1,p);
        }
    }
}
void print(int se){
//  for (int i=1;i<=k;++i) printf("%d ",link[i]);printf("dd ");
//  for (int i=1;i<=k;++i) printf("%d ",b[se][i]);printf("dd ");
//  for (int i=1;i<=k;++i) printf("%d ",flag[b[se][i]]);
    memset(father,0,sizeof(father));
    int num=0;int flagf[60];
    memset(flagf,0,sizeof(flagf));
    for (int i=1;i<=k;++i) father[i]=b[se][i];
    bool flag1[10];memcpy(flag1,flag,sizeof(flag));
    for (int i=1;i<=k;++i){
        if (flag1[b[se][i]]==true){
            father[k+1]=k+1;
            for (int j=1;j<=k;++j){
                if (b[se][j]==b[se][i]) father[j]=k+1;
            }       
            flag1[b[se][i]]=false;
            //if (link[i]==1) flagf[k+1]=num;
        }
    }
    //for (int i=1;i<=k+1;++i) printf("%d ",father[i]);
    int sheet[10];memset(sheet,-1,sizeof(sheet));
    //printf("\n");
    for (int i=2;i<=k+1;++i) {
        if (sheet[father[i]]==-1){
            ++num;flagf[i]=num;sheet[father[i]]=num;continue;
        }
        flagf[i]=sheet[father[i]];
    }
    int tmp=0;
    //for (int i=1;i<=k+1;++i) printf("%d ",flagf[i]);
    for (int i=2;i<=k+1;++i) tmp=tmp*10+flagf[i];
//  printf("%d\n",tmp);
    map[se][sh[tmp]]+=1;
}
void dfs2(int se,int position){

    if (position==k+1){
        print(se);return;
    }
    if (position==1){
        bool tmp=false;
        for (int i=2;i<=k;++i) if (b[se][i]==1) {
            tmp=true;break;
        }
        if (tmp==false){
            flag[1]=true;link[1]=1;
            dfs2(se,position+1);
            return;
        }

    }
    if (flag[b[se][position]]){
        link[position]=0;numm[b[se][position]]++;//printf("%d ",numm[1]);
        dfs2(se,position+1);
        if (numm[b[se][position]]) numm[b[se][position]]--;
        return; 
    }
    flag[b[se][position]]=true;link[position]=1;numm[b[se][position]]++;//printf("%d ",numm[1]);
    dfs2(se,position+1);    
    if (numm[b[se][position]]) numm[b[se][position]]--;
    if (numm[b[se][position]]==0) flag[b[se][position]]=false;link[position]=0;
    dfs2(se,position+1);
}
void build(){
    memset(base1.f,0,sizeof(base1.f));
    for (int i=1;i<=num;++i) {
        for (int j=1;j<=num;++j) if (i==j) base1.f[i][j]=1;
    }
    base1.l=base1.c=num;
}
int main(){
    freopen("count.in","r",stdin);
    freopen("count.out","w",stdout);
    scanf("%d %lld",&k,&n);num=0;
    dfs(1,1);
    //printf("%d ",num);
    for (int i=1;i<=num;++i) 
        for (int j=1;j<=k;++j) s[i]=s[i]*10+b[i][j];
//  sort(s+1,s+num+1);
//  printf("%d\n",s[6]);
    //for (int i=1;i<=num;++i) printf("%d ",s[i]);
    for (int i=1;i<=num;++i) sh[s[i]]=i;
//  dfs2(2,1);  
    int tmp1[60];memset(tmp1,0,sizeof(tmp1));
    for (int i=1;i<=num;++i){
        int t=s[i],p=0;
        while (t>0) a[++p]=t%10,t/=10;
        for (int j=1;j<=k;++j){
            int p=1;
            for (int i1=1;i1<=k;++i1){
                if (i1!=j&&a[i1]==a[j]) ++p;
            }
            if (tmp1[i]<p) tmp1[i]=p;
        }
        memset(flag,false,sizeof(flag));memset(link,0,sizeof(link));
        memset(numm,0,sizeof(numm));
        dfs2(i,1); //printf("asdfasdf\n");
    }
//  for (int i=1;i<=num;++i) printf("%d ",NUM[tmp1[i]]);
    //print(8);
    /*for (int i=1;i<=num;++i){
        for (int j=1;j<=num;++j) printf("%d ",map[i][j]);printf("\n");
    }*/
    build();
    matrix p;p.l=num;p.c=num;
    for (int i=1;i<=num;++i)
        for (int j=1;j<=num;++j) p.f[i][j]=map[i][j];
/*  for (int i=1;i<=num;++i){
        for (int j=1;j<=num;++j) printf("%d ",base1.f[i][j]);printf("\n");
    }   */
    p=pow(p,n-k);
    /*for (int i=1;i<=num;++i){
        for (int j=1;j<=num;++j) printf("%d ",p.f[i][j]);
        printf("\n");
    }*/
    long long ans=0;;
    for (int i=1;i<=num;++i) (ans+=p.f[i][1]*NUM[tmp1[i]])%=N;
    printf("%lld",ans);
    /*for (int i=1;i<=n;++i){
        for (int j=1;j<=n;++j){
            if (i==j) {mat[i][j]=i-max(1,i-k)+min(n,k+i)-i;continue;}//前面多少连过来+后面多少连过去 
            if (abs(i-j)<=k) {mat[i][j]=-1;continue;}
            mat[i][j]=0;
        }
    }
    n--;
    memset(get,false,sizeof(get));*/
    return 0;
}