Score : $500$ points

问题描述

你给定一个简单的有向图，包含 $N$ 个顶点，编号为 $1$ 到 $N$，以及 $M$ 条边，编号为 $1$ 到 $M$。其中第 $i$ 条边是从顶点 $u_i$ 到顶点 $v_i$ 的有向边。

你可以执行以下操作零次或多次：

• 选择两个不同的顶点 $x$ 和 $y$，确保当前不存在从顶点 $x$ 到顶点 $y$ 的有向边，然后添加一条从顶点 $x$ 到顶点 $y$ 的有向边。

找出需要执行上述操作的最小次数，使得图满足以下条件：

• 对于每三个不同的顶点 $a$、$b$ 和 $c$，如果存在从顶点 $a$ 到顶点 $b$ 以及从顶点 $b$ 到顶点 $c$ 的有向边，则也必须存在从顶点 $a$ 到顶点 $c$ 的有向边。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given a simple directed graph with $N$ vertices numbered $1$ to $N$ and $M$ edges numbered $1$ to $M$. Edge $i$ is a directed edge from vertex $u_i$ to vertex $v_i$.

You may perform the following operation zero or more times.

• Choose distinct vertices $x$ and $y$ such that there is no directed edge from vertex $x$ to vertex $y$, and add a directed edge from vertex $x$ to vertex $y$.

Find the minimum number of times you need to perform the operation to make the graph satisfy the following condition.

• For every triple of distinct vertices $a$, $b$, and $c$, if there are directed edges from vertex $a$ to vertex $b$ and from vertex $b$ to vertex $c$, there is also a directed edge from vertex $a$ to vertex $c$.

Constraints

• $3 \leq N \leq 2000$
• $0 \leq M \leq 2000$
• $1 \leq u_i ,v_i \leq N$
• $u_i \neq v_i$
• $(u_i,v_i) \neq (u_j,v_j)$ if $i \neq j$.
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$ $M$

$u_1$ $v_1$

$\vdots$

$u_M$ $v_M$

Output

Print the answer.

Sample Input 1

4 3
2 4
3 1
4 3

Sample Output 1

3

Initially, the condition is not satisfied because, for instance, for vertices $2$, $4$, and $3$, there are directed edges from vertex $2$ to vertex $4$ and from vertex $4$ to vertex $3$, but not from vertex $2$ to vertex $3$.

You can make the graph satisfy the condition by adding the following three directed edges:

• one from vertex $2$ to vertex $3$,
• one from vertex $2$ to vertex $1$, and
• one from vertex $4$ to vertex $1$.

On the other hand, the condition cannot be satisfied by adding two or fewer edges, so the answer is $3$.

Sample Input 2

292 0

Sample Output 2

0

Sample Input 3

5 8
1 2
2 1
1 3
3 1
1 4
4 1
1 5
5 1

Sample Output 3

12

update @ 2024/3/10 12:11:09