【CF Round #545】C. Museums Tour




\(n,m\leq 10^5, d\leq 50\)


input1 output1
4 5 3
3 1
1 2
2 4
4 1
2 3
input2 output2
3 3 7
1 2
1 3
2 3





假设第\(i\)个点星期\(j\)代表的点为\((i, j)\),如果存在路径\((i, j) \rightarrow (i, j’) (j \neq j’) \),则设\(\Delta T=(j’-j+d)\%d \),必同时存在路径\((i, k) \rightarrow (i, (k + \Delta T)\%d) \), \((k\in (0, d – 1))\),因此,若\( j’ + k\Delta T \equiv j (\mathbf {mod}\;d)\)有正整数解,则同时存在路径\((i,j’) \rightarrow (i, j)\),又因为\(j + \Delta T \equiv j’ (\mathbf {mod}\;d)\),带入原方程可化为\((k + 1)\Delta T \equiv 0 (\mathbf {mod} \; d)\), 显然存在正整数解\(k=2d-1\),因此\((i,j),(i,j’)\),在同一个SCC中。




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