Zoj 2344 Toral Tickets (数学_Polya)
题目链接：http://acm.zju.edu.cn/onlinejudge/showProblem.do?problemId=1399
题目大意：给定一张有n*m个格子的纸，每个格子有黑白两种颜色可以染。现在先将纸按长边粘起来得到一个圆柱，再将纸按短边拈起来得到一个游泳圈。如果两种染色方案卷起来后是一样的，那么它们同构。问不同构的的染色方案。n,m <= 20.
解题思路：Polya的核心是找置换，本题有两类置换，一类是滚动，一类是旋转。
我们将n*m的纸想成一个矩阵，那么第一行是1,2,3...m,第二行是m+1...m*2，以此类推。
第一类滚动置换有n*m个。行和列是分开的，行的滚动有n个，列的滚动有m个，它们相乘就是总置换数。每个置换的循环节个数我们就模拟下，O（n*m）找到循环节。
第二类旋转置换要分类讨论，当n！= m时，我们可以将整个矩阵旋转180度，这里是旋转而不是翻转，我写了Rorate函数将矩阵旋转九十度，Rorate两次.当n==m时，不仅可以旋转180度，旋转90度和270度都要考虑在内。
也就是说当n！=m时，置换有2*n*m个，当n==m时，置换有4*n*m个，再套个高精度模板就AC了。n和m很小，随便模拟。
ac后我很ws地把结果打成表拿去提交，时间就是0s...
高精度模板太长，我就不发上来了，百度下一坨。
测试数据：
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C艹代码：
int n,m,arr[MAX][MAX],vis[MAX][MAX];
int cnt[MAX*MAX],temp[MAX][MAX];
int Calculate(int arr[][MAX]) {
int i,j,k,tp,ii,jj,tot = 0;
memset(vis,0,sizeof(vis));
for (i = 1; i <= n; ++i)
for (j = 1; j <= m; ++j)
if (vis[i][j] == 0) {
tot++;
ii = i,jj = j;
// printf("tot = %d\n",tot);
while (1) {
vis[ii][jj] = 1;
tp = arr[ii][jj];
//printf("ii%d jj%d tp%d\n",ii,jj,tp);
ii = (tp + m - 1) / m;
jj = tp - (ii - 1) * m;
if (vis[ii][jj]) break;
}
}
return tot;
}
void UpMove(int arr[][MAX]) {
memcpy(arr[n+1],arr[1],sizeof(arr[1]));
for (int i = 2; i <= n; ++i)
memcpy(arr[i-1],arr[i],sizeof(arr[i]));
memcpy(arr[n],arr[n+1],sizeof(arr[n+1]));
}
void LeftMove(int arr[][MAX]) {
int i,j,k;
for (i = 1; i <= n; ++i)
arr[i][m+1] = arr[i][1];
for (j = 2; j <= m; ++j)
for (i = 1; i <= n; ++i)
arr[i][j-1] = arr[i][j];
for (i = 1; i <= n; ++i)
arr[i][m] = arr[i][1+m];
}
void Rotate(int arr[][MAX]) {
int i,j,k,brr[MAX][MAX];
for (i = 1; i <= n; ++i)
for (j = 1; j <= m; ++j)
brr[j][n+1-i] = arr[i][j];
memcpy(arr,brr,sizeof(brr));
k = n,n = m, m = k;
}
void Print(int arr[][MAX]) {
for (int i = 1; i <= n; ++i)
for (int j = 1; j <= m; ++j)
printf("%d%c",arr[i][j],j==m?'\n':' ');
}
BigNum Power(int n,int k) {
BigNum x = n,ans = 1;
while (k) {
if (k & 1) ans = x * ans;
x = x * x,k >>= 1;
}
return ans;
}
int main()
{
int i,j,k,ii,jj,tp;
while (scanf("%d%d",&n,&m) != EOF) {
if (n > m) k = n,n = m,m = k;
memset(arr,0,sizeof(arr));
memset(cnt,0,sizeof(cnt));
for (i = 1; i <= n; ++i)
for (j = 1; j <= m; ++j)
arr[i][j] = j + (i - 1) * m;
for (ii = 1; ii <= n; ++ii) {
memcpy(temp,arr,sizeof(arr));
for (jj = 1; jj <= m; ++jj) {
tp = Calculate(temp); //0度
cnt[tp]++;
Rotate(temp); //90度
if (n == m)
tp = Calculate(temp),cnt[tp]++;
Rotate(temp); //180度
tp = Calculate(temp);
cnt[tp]++;
Rotate(temp); //270度
if (n == m)
tp = Calculate(temp),cnt[tp]++;
Rotate(temp); //360度
LeftMove(temp);
}
UpMove(arr);
}
BigNum ans = 0;
for (i = 1; i <= n * m; ++i)
if (cnt[i]) ans = ans + Power(2,i) * cnt[i];
if (n != m) ans = ans / (2 * n * m);
else ans = ans / (4 * n * m);
ans.print();
}
}
本文ZeroClock原创，但可以转载，因为我们是兄弟。