codeforces#P1674G. Remove Directed Edges
Remove Directed Edges
Description
You are given a directed acyclic graph, consisting of $n$ vertices and $m$ edges. The vertices are numbered from $1$ to $n$. There are no multiple edges and self-loops.
Let $\mathit{in}_v$ be the number of incoming edges (indegree) and $\mathit{out}_v$ be the number of outgoing edges (outdegree) of vertex $v$.
You are asked to remove some edges from the graph. Let the new degrees be $\mathit{in'}_v$ and $\mathit{out'}_v$.
You are only allowed to remove the edges if the following conditions hold for every vertex $v$:
- $\mathit{in'}_v < \mathit{in}_v$ or $\mathit{in'}_v = \mathit{in}_v = 0$;
- $\mathit{out'}_v < \mathit{out}_v$ or $\mathit{out'}_v = \mathit{out}_v = 0$.
Let's call a set of vertices $S$ cute if for each pair of vertices $v$ and $u$ ($v \neq u$) such that $v \in S$ and $u \in S$, there exists a path either from $v$ to $u$ or from $u$ to $v$ over the non-removed edges.
What is the maximum possible size of a cute set $S$ after you remove some edges from the graph and both indegrees and outdegrees of all vertices either decrease or remain equal to $0$?
The first line contains two integers $n$ and $m$ ($1 \le n \le 2 \cdot 10^5$; $0 \le m \le 2 \cdot 10^5$) — the number of vertices and the number of edges of the graph.
Each of the next $m$ lines contains two integers $v$ and $u$ ($1 \le v, u \le n$; $v \neq u$) — the description of an edge.
The given edges form a valid directed acyclic graph. There are no multiple edges.
Print a single integer — the maximum possible size of a cute set $S$ after you remove some edges from the graph and both indegrees and outdegrees of all vertices either decrease or remain equal to $0$.
Input
The first line contains two integers $n$ and $m$ ($1 \le n \le 2 \cdot 10^5$; $0 \le m \le 2 \cdot 10^5$) — the number of vertices and the number of edges of the graph.
Each of the next $m$ lines contains two integers $v$ and $u$ ($1 \le v, u \le n$; $v \neq u$) — the description of an edge.
The given edges form a valid directed acyclic graph. There are no multiple edges.
Output
Print a single integer — the maximum possible size of a cute set $S$ after you remove some edges from the graph and both indegrees and outdegrees of all vertices either decrease or remain equal to $0$.
Samples
3 3
1 2
2 3
1 3
2
5 0
1
7 8
7 1
1 3
6 2
2 3
7 2
2 4
7 3
6 3
3
Note
In the first example, you can remove edges $(1, 2)$ and $(2, 3)$. $\mathit{in} = [0, 1, 2]$, $\mathit{out} = [2, 1, 0]$. $\mathit{in'} = [0, 0, 1]$, $\mathit{out'} = [1, 0, 0]$. You can see that for all $v$ the conditions hold. The maximum cute set $S$ is formed by vertices $1$ and $3$. They are still connected directly by an edge, so there is a path between them.
In the second example, there are no edges. Since all $\mathit{in}_v$ and $\mathit{out}_v$ are equal to $0$, leaving a graph with zero edges is allowed. There are $5$ cute sets, each contains a single vertex. Thus, the maximum size is $1$.
In the third example, you can remove edges $(7, 1)$, $(2, 4)$, $(1, 3)$ and $(6, 2)$. The maximum cute set will be $S = \{7, 3, 2\}$. You can remove edge $(7, 3)$ as well, and the answer won't change.
Here is the picture of the graph from the third example:
