U208746 [F] Graph Theory

Bobo has an **undirected** graph $G$ with $n$ vertices labeled by $1, \dots, n$ and $n$ edges. For each $1 \leq i \leq n$, there is an edge between the vertex $i$ and the vertex $(i \bmod n) + 1$. He also has a list of $m$ pairs $(a_1, b_1), \dots, (a_m, b_m)$. Now, Bobo is going to choose an $i$ and remove the edge between the vertex $i$ and the vertex $(i \bmod n) + 1$. Let $\delta_i(u, v)$ be the number of edges on the shortest path between the $u$-th and the $v$-th vertex **after the removal**. Choose an $i$ to minimize the maximum among $\delta_i(a_1, b_1), \dots, \delta_i(a_m, b_m)$. Formally, find the value of $$ \min_{1 \leq i \leq n}\left\{\max_{1 \leq j \leq m} \delta_i(a_j, b_j)\right\}. $$

The input consists of several test cases terminated by end-of-file. For each test case, The first line contains two integers $n$ and $m$. For the following $m$ lines, the $i$-th line contains two integers $a_i$ and $b_i$. * $2 \leq n \leq 2 \times 10^5$ * $1 \leq m \leq 2 \times 10^5$ * $1 \leq a_i, b_i \leq n$ for each $1 \leq i \leq m$ * In each input, the sum of $n$ does not exeed $2 \times 10^5$. The sum of $m$ does not exceed $2 \times 10^5$.

For each test case, output an integer which denotes the minimum value.

For the first case, | $i$ | $\delta_i(1, 2)$ | $\delta_i(2, 3)$ | | ---- | ---------------- | ---------------- | | $1$ | $2$ | $1$ | | $2$ | $1$ | $2$ | | $3$ | $1$ | $1$ | Choosing $i = 3$ yields the minimum value $1$.