标签:scanf 裸题 getc eof const span space nbsp using
原文链接https://www.cnblogs.com/zhouzhendong/p/51Nod1518.html
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首先,我们忽略那个“稳定”的要求,求方案数。
显然是一个插头dp裸题,我们可以在 $O(n\cdot 2^n)$ 的时间复杂度中求出所有长宽的矩形区域的覆盖方案数。
然后我们考虑容斥原理,奇加偶减。首先,枚举那些相邻行之间有一条不穿过骨牌的直线,然后,用一个 $O(n)$ DP 来解决相邻列之间分割线的容斥。
总的时间复杂度 $O(n^22^n)$ 。打出表之后,询问 $O(1)$ 。
看着那些运行效率榜上15MS的代码我表示不服。于是交了一份 0MS 的代码。正常的代码在这份代码之后。
#include <bits/stdc++.h> int n,m,ans[17][17]={ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,6,0,108,0,1182,0,10338,0,79818,0,570342}, {0,0,0,0,0,6,0,124,62,1646,1630,18120,25654,180288,317338,1684956,3416994}, {0,0,0,0,0,0,124,0,13514,0,765182,0,32046702,0,136189727,0,378354090}, {0,0,0,0,0,108,62,13514,25506,991186,3103578,57718190,238225406,965022920,388537910,937145938,315565230}, {0,0,0,0,0,0,1646,0,991186,0,262834138,0,462717719,0,560132342,0,699538539}, {0,0,0,0,0,1182,1630,765182,3103578,262834138,759280991,264577134,712492587,886997066,577689269,510014880,807555438}, {0,0,0,0,0,0,18120,0,57718190,0,264577134,0,759141342,0,567660301,0,47051173}, {0,0,0,0,0,10338,25654,32046702,238225406,462717719,712492587,759141342,398579168,83006813,821419653,942235780,558077885}, {0,0,0,0,0,0,180288,0,965022920,0,886997066,0,83006813,0,690415372,0,620388364}, {0,0,0,0,0,79818,317338,136189727,388537910,560132342,577689269,567660301,821419653,690415372,796514774,696587391,175421667}, {0,0,0,0,0,0,1684956,0,937145938,0,510014880,0,942235780,0,696587391,0,856463275}, {0,0,0,0,0,570342,3416994,378354090,315565230,699538539,807555438,47051173,558077885,620388364,175421667,856463275,341279366} }; int main(){ while (~scanf("%d%d",&n,&m)) printf("%d\n",ans[n][m]); return 0; }
正常的代码
#include <bits/stdc++.h> using namespace std; int read(){ int x=0; char ch=getchar(); while (!isdigit(ch)) ch=getchar(); while (isdigit(ch)) x=(x<<1)+(x<<3)+ch-48,ch=getchar(); return x; } const int N=17,S=1<<16,mod=1e9+7; int n,m,dp[2][S],tot[N][N],ans[N][N]; int gbit(int v,int d){ return (v>>(d-1))&1; } void Solve_tot(int n,int m){ memset(dp,0,sizeof dp); int T0=1,T1=0; dp[T1][(1<<m)-1]=1; for (int i=1;i<=n;i++){ for (int j=1;j<=m;j++){ T0^=1,T1^=1; memset(dp[T1],0,sizeof dp[T1]); for (int s=0;s<(1<<m);s++){ int v=dp[T0][s]; if (!v) continue; dp[T1][s^(1<<(j-1))]=(dp[T1][s^(1<<(j-1))]+v)%mod; if (j>1&&!gbit(s,j-1)&&gbit(s,j)){ int _s=s^(1<<(j-2)); dp[T1][_s]=(dp[T1][_s]+v)%mod; } } } tot[i][m]=dp[T1][(1<<m)-1]; } } void Get_tot(int n){ for (int m=1;m<=16;m++) Solve_tot(n,m); } int GetV(int n,int s,int len){ int v=1; for (int i=1,j;i<=n;i=j){ for (j=i;j<n&&!((s>>j)&1);j++); j++; v=1LL*v*tot[j-i][len]%mod; } return v; } int cnt_1(int v){ int ans=0; while (v) ans+=v&1,v>>=1; return ans; } void Solve_ans(int n,int m){ int dp[N],v[N]; for (int s=0;s<(1<<n);s++){ if (!(s&1)) continue; memset(dp,0,sizeof dp); for (int i=1;i<=m;i++) v[i]=GetV(n,s,i); dp[0]=1; for (int i=1;i<=m;i++) for (int j=0;j<i;j++) dp[i]=(-1LL*dp[j]*v[i-j]+dp[i])%mod; int f=(cnt_1(s)&1)?-1:1; for (int i=1;i<=m;i++) ans[n][i]=(ans[n][i]+f*dp[i])%mod; } } void Get_ans(int m){ memset(ans,0,sizeof ans); for (int n=1;n<=16;n++) Solve_ans(n,m); for (int i=1;i<=m;i++) for (int j=1;j<=m;j++) ans[i][j]=(ans[i][j]+mod)%mod; } int main(){ Get_tot(16); Get_ans(16); while (~scanf("%d%d",&n,&m)) printf("%d\n",ans[n][m]); return 0; }
51Nod1518 稳定多米诺覆盖 动态规划 插头dp 容斥原理
标签:scanf 裸题 getc eof const span space nbsp using
原文地址:https://www.cnblogs.com/zhouzhendong/p/51Nod1518.html