51nod1446 Kirchhoff矩阵+Gauss消元+容斥+折半DFS

Posted 2020-09-23 SiriusRen

思路：

//By SiriusRen
#include <cstdio>
#include <cstring>
#include <algorithm>
using namespace std;
typedef long long ll;
const int mod=1000000007;
int cases,n,maxval,a[44][44],C[44][44],f[44],val[44],X,g[44];
int top2,top,bgn,half,cnts2[22],allnum[44],Ans,T;
struct Node{int wei,num;Node(){}Node(int x,int y){wei=x,num=y;}}s[5+(1<<20)],s2[5+(1<<20)];
bool operator<(Node a,Node b){return a.wei<b.wei;}
int pow(ll x,int y){
    ll res=1;
    while(y){
        if(y&1)res=res*x%mod;
        x=x*x%mod,y>>=1;
    }return (int)res;
}
int Gauss(int n){
    int f=1;
    for(int i=1;i<=n;i++){
        int j=i;while(!a[i][j]&&j<=n)j++;
        if(j==n+1)return 0;
        if(j!=i){
            for(int k=1;k<=n;k++)swap(a[i][k],a[j][k]);
            f*=-1;
        }
        int t=pow(a[i][i],mod-2);
        for(int j=i+1;j<=n;j++){
            int ww=1ll*a[j][i]*t%mod;
            for(int k=i;k<=n;k++)a[j][k]=(a[j][k]-1ll*ww*a[i][k]%mod+mod)%mod;
        }
    }
    ll ans=1;
    for(int i=1;i<=n;i++)ans=ans*a[i][i]%mod;
    if(f==-1)ans=(mod-ans)%mod;
    return ans;
}
void Kirchhoff(int x){
    memset(a,0,sizeof(a));
    for(int i=1;i<=n;i++)
        if(i<=x)for(int j=max(i+1,X+1);j<=n;j++)a[i][j]--,a[j][i]--,a[i][i]++,a[j][j]++;
        else for(int j=i+1;j<=n;j++)a[i][j]--,a[j][i]--,a[i][i]++,a[j][j]++;
}
bool cmp(int x,int y){return x>y;}
void dfs(int x,int wei,int deep){
    !T?s[top++]=Node(wei,deep):s2[top2++]=Node(wei,deep);
    for(int i=x;i>bgn;i--)if(wei+val[i]<=maxval)dfs(i-1,wei+val[i],deep+1);
}
signed main(){
    for(int i=0;i<=40;i++){
        C[i][0]=C[i][i]=1;
        for(int j=1;j<i;j++)C[i][j]=(C[i-1][j-1]+C[i-1][j])%mod;
    }
    scanf("%d",&cases);
    while(cases--){
        memset(allnum,0,sizeof(allnum));
        memset(cnts2,0,sizeof(cnts2));
        top=top2=X=bgn=Ans=T=0;
        scanf("%d%d",&n,&maxval);
        for(int i=1;i<=n;i++){
            scanf("%d",&val[i]);
            if(~val[i])X++;
        }half=(X+1)/2;
        for(int i=0;i<=X;i++)Kirchhoff(i),f[i]=g[i]=Gauss(n-1);
        for(int i=X;~i;i--)
            for(int j=i+1;j<=X;j++)f[i]=((f[i]-1ll*C[X-i][j-i]*f[j])%mod+mod)%mod;
        sort(val+1,val+1+n,cmp),random_shuffle(val+1,val+1+X);
        dfs(half,0,0),sort(s,s+top);
        T=1,bgn=half,dfs(X,0,0),sort(s2,s2+top2);
        for(int i=0;i<top2;i++)cnts2[s2[i].num]++;
        for(int i=0,j=top2-1;i<top;i++){
            while(s[i].wei+s2[j].wei>maxval)cnts2[s2[j].num]--,j--;
            for(int k=0;k<=X-half;k++)(allnum[k+s[i].num]+=cnts2[k])%=mod;
        }
        for(int i=0;i<=X;i++)Ans=(Ans+1ll*f[X-i]*allnum[i])%mod;
        printf("%d\\n",(Ans+mod)%mod);
    }
}