[cf963E]Circles of Waiting
将与原点距离大于$R$的点缩为一个点$t$,即终点
做法1
定义$f_{i}$表示从$i$到$t$的期望步数,即$f_{i}=\begin{cases}\sum_{(i,j)\in E}w_{(i,j)}f_{j}+1&(j\ne t)\\0&(j=t)\end{cases}$
直接对其高斯消元,时间复杂度为$o(n^{3}m^{3})$
做法2
事实上,如果将其按照顺序从上到下、从左到右依次编号,并依次消元,编号为$x$的行在(消元的)任意时刻,都仅有$[x-m,x+m]$这些列的元素非0
由此,显然在消元时,仅需要枚举其后$m$列,且每一列仅需枚举$o(m)$个元素,即做到$o(n^{2}m^{2})$的复杂度
做法3
对于所有元素,若其上方的格子不存在,即将其作为一个变量,否则通过其上方的格子的方程即可确定其的表示,最终即仅有$o(m)$个变量,以及下方不存在格子的位置未保证其成立
在$o(nm^{2})$的时间内推出此式子,并对$m$元解方程,复杂度为$o(m^{3})$
类似地,也可以从左到右去枚举,做到$o(n^{2}m)$的复杂度
综上,即可做到$o(nm\min(n,m)))$的复杂度
![](https://images.cnblogs.com/OutliningIndicators/ContractedBlock.gif)
1 #include<bits/stdc++.h> 2 using namespace std; 3 #define N 105 4 #define M 8005 5 #define mod 1000000007 6 int V,R,a,b,c,d,id[N][N],A[M][M],ans[M]; 7 int pow(int n,int m){ 8 int s=n,ans=1; 9 while (m){ 10 if (m&1)ans=1LL*ans*s%mod; 11 s=1LL*s*s%mod; 12 m>>=1; 13 } 14 return ans; 15 } 16 void guess(){ 17 for(int i=1;i<=V;i++){ 18 int x=pow(A[i][i],mod-2); 19 for(int j=i;j<=V+1;j++)A[i][j]=1LL*A[i][j]*x%mod; 20 for(int j=i+1;j<=min(i+2*R,V);j++){ 21 if (A[j][i]){ 22 int x=A[j][i]; 23 for(int k=i;k<=min(i+2*R,V);k++)A[j][k]=(A[j][k]-1LL*A[i][k]*x%mod+mod)%mod; 24 A[j][V+1]=(A[j][V+1]-1LL*A[i][V+1]*x%mod+mod)%mod; 25 } 26 } 27 } 28 for(int i=V;i;i--){ 29 for(int j=i+1;j<=V;j++)A[i][V+1]=(A[i][V+1]-1LL*A[i][j]*ans[j]%mod+mod)%mod; 30 ans[i]=A[i][V+1]; 31 } 32 } 33 int main(){ 34 scanf("%d%d%d%d%d",&R,&a,&b,&c,&d); 35 int inv=pow(a+b+c+d,mod-2); 36 a=1LL*a*inv%mod,b=1LL*b*inv%mod,c=1LL*c*inv%mod,d=1LL*d*inv%mod; 37 for(int i=-R;i<=R;i++) 38 for(int j=-R;j<=R;j++) 39 if (i*i+j*j<=R*R)id[i+R][j+R]=++V; 40 for(int i=-R;i<=R;i++) 41 for(int j=-R;j<=R;j++){ 42 if (i*i+j*j<=R*R){ 43 int ii=i+R,jj=j+R; 44 if ((i-1)*(i-1)+j*j<=R*R)A[id[ii][jj]][id[ii-1][jj]]=(A[id[ii][jj]][id[ii-1][jj]]+mod-a)%mod; 45 if (i*i+(j-1)*(j-1)<=R*R)A[id[ii][jj]][id[ii][jj-1]]=(A[id[ii][jj]][id[ii][jj-1]]+mod-b)%mod; 46 if ((i+1)*(i+1)+j*j<=R*R)A[id[ii][jj]][id[ii+1][jj]]=(A[id[ii][jj]][id[ii+1][jj]]+mod-c)%mod; 47 if (i*i+(j+1)*(j+1)<=R*R)A[id[ii][jj]][id[ii][jj+1]]=(A[id[ii][jj]][id[ii][jj+1]]+mod-d)%mod; 48 } 49 } 50 for(int i=1;i<=V;i++)A[i][i]=A[i][V+1]=1; 51 guess(); 52 printf("%d",ans[id[R][R]]); 53 }
![](https://images.cnblogs.com/OutliningIndicators/ContractedBlock.gif)
1 #include<bits/stdc++.h> 2 using namespace std; 3 #define N 105 4 #define M 8005 5 #define mod 1000000007 6 int V,VV,R,a,b,c,d,sum,id[N][N],g[M][N<<1],A[M][N<<1],ans[M]; 7 int pow(int n,int m){ 8 int s=n,ans=1; 9 while (m){ 10 if (m&1)ans=1LL*ans*s%mod; 11 s=1LL*s*s%mod; 12 m>>=1; 13 } 14 return ans; 15 } 16 void guess(){ 17 for(int i=1;i<=V;i++){ 18 int k=-1; 19 for(int j=i;j<=V;j++) 20 if (A[j][i]){ 21 k=j; 22 break; 23 } 24 if (k!=i){ 25 for(int j=i;j<=V+1;j++)swap(A[i][j],A[k][j]); 26 } 27 int x=pow(A[i][i],mod-2); 28 for(int j=i;j<=V+1;j++)A[i][j]=1LL*A[i][j]*x%mod; 29 for(int j=i+1;j<=V;j++) 30 if (A[j][i]){ 31 int x=A[j][i]; 32 for(int k=i;k<=V+1;k++)A[j][k]=(A[j][k]-1LL*A[i][k]*x%mod+mod)%mod; 33 } 34 } 35 for(int i=V;i;i--){ 36 for(int j=i+1;j<=V;j++)A[i][V+1]=(A[i][V+1]-1LL*A[i][j]*ans[j]%mod+mod)%mod; 37 ans[i]=A[i][V+1]; 38 } 39 } 40 int main(){ 41 scanf("%d%d%d%d%d",&R,&a,&b,&c,&d); 42 int inv=pow(a+b+c+d,mod-2); 43 a=1LL*a*inv%mod,b=1LL*b*inv%mod,c=1LL*c*inv%mod,d=1LL*d*inv%mod; 44 for(int i=-R;i<=R;i++) 45 for(int j=-R;j<=R;j++) 46 if (i*i+j*j<=R*R)id[i+R][j+R]=++V; 47 V=0; 48 for(int i=-R;i<=R;i++) 49 for(int j=-R;j<=R;j++) 50 if ((i*i+j*j<=R*R)&&((i-1)*(i-1)+j*j>R*R))g[id[i+R][j+R]][++V]=1; 51 for(int i=-R;i<=R;i++) 52 for(int j=-R;j<=R;j++) 53 if ((i*i+j*j<=R*R)&&((i-1)*(i-1)+j*j<=R*R)){ 54 int ii=i+R,jj=j+R; 55 memcpy(g[id[ii][jj]],g[id[ii-1][jj]],sizeof(g[id[ii][jj]])); 56 g[id[ii][jj]][V+1]=(g[id[ii][jj]][V+1]+mod-1)%mod; 57 if ((i-2)*(i-2)+j*j<=R*R){ 58 for(int k=1;k<=V+1;k++)g[id[ii][jj]][k]=(g[id[ii][jj]][k]-1LL*a*g[id[ii-2][jj]][k]%mod+mod)%mod; 59 } 60 if ((i-1)*(i-1)+(j-1)*(j-1)<=R*R){ 61 for(int k=1;k<=V+1;k++)g[id[ii][jj]][k]=(g[id[ii][jj]][k]-1LL*b*g[id[ii-1][jj-1]][k]%mod+mod)%mod; 62 } 63 if ((i-1)*(i-1)+(j+1)*(j+1)<=R*R){ 64 for(int k=1;k<=V+1;k++)g[id[ii][jj]][k]=(g[id[ii][jj]][k]-1LL*d*g[id[ii-1][jj+1]][k]%mod+mod)%mod; 65 } 66 int x=pow(c,mod-2); 67 for(int k=1;k<=V+1;k++)g[id[ii][jj]][k]=1LL*g[id[ii][jj]][k]*x%mod; 68 } 69 for(int i=-R;i<=R;i++) 70 for(int j=-R;j<=R;j++) 71 if ((i*i+j*j<=R*R)&&((i+1)*(i+1)+j*j>R*R)){ 72 int ii=i+R,jj=j+R; 73 memcpy(A[++VV],g[id[ii][jj]],sizeof(g[id[ii][jj]])); 74 if ((i-1)*(i-1)+j*j<=R*R){ 75 for(int k=1;k<=V+1;k++)A[VV][k]=(A[VV][k]-1LL*a*g[id[ii-1][jj]][k]%mod+mod)%mod; 76 } 77 if (i*i+(j-1)*(j-1)<=R*R){ 78 for(int k=1;k<=V+1;k++)A[VV][k]=(A[VV][k]-1LL*b*g[id[ii][jj-1]][k]%mod+mod)%mod; 79 } 80 if ((i+1)*(i+1)+j*j<=R*R){ 81 for(int k=1;k<=V+1;k++)A[VV][k]=(A[VV][k]-1LL*c*g[id[ii+1][jj]][k]%mod+mod)%mod; 82 } 83 if (i*i+(j+1)*(j+1)<=R*R){ 84 for(int k=1;k<=V+1;k++)A[VV][k]=(A[VV][k]-1LL*d*g[id[ii][jj+1]][k]%mod+mod)%mod; 85 } 86 A[VV][V+1]=mod+1-A[VV][V+1]; 87 } 88 guess(); 89 sum=g[id[R][R]][V+1]; 90 for(int i=1;i<=V;i++)sum=(sum+1LL*g[id[R][R]][i]*ans[i])%mod; 91 printf("%d",sum); 92 }