The Number of Imposters
Contest: Codeforces Round 747 (Div. 2)
Difficulty rating: 1700
Tags: constructive algorithms, dfs and similar, dp, dsu, graphs

Theofanis started playing the new online game called "Among them". However, he always plays with Cypriot players, and they all have the same name: "Andreas" (the most common name in Cyprus).

In each game, Theofanis plays with $$$n$$$ other players. Since they all have the same name, they are numbered from $$$1$$$ to $$$n$$$.

The players write $$$m$$$ comments in the chat. A comment has the structure of "$$$i$$$ $$$j$$$ $$$c$$$" where $$$i$$$ and $$$j$$$ are two distinct integers and $$$c$$$ is a string ($$$1 \le i, j \le n$$$; $$$i \neq j$$$; $$$c$$$ is either imposter or crewmate). The comment means that player $$$i$$$ said that player $$$j$$$ has the role $$$c$$$.

An imposter always lies, and a crewmate always tells the truth.

Help Theofanis find the maximum possible number of imposters among all the other Cypriot players, or determine that the comments contradict each other (see the notes for further explanation).

Note that each player has exactly one role: either imposter or crewmate.

Input format:
The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. Description of each test case follows.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le n \le 2 \cdot 10^5$$$; $$$0 \le m \le 5 \cdot 10^5$$$) — the number of players except Theofanis and the number of comments.

Each of the next $$$m$$$ lines contains a comment made by the players of the structure "$$$i$$$ $$$j$$$ $$$c$$$" where $$$i$$$ and $$$j$$$ are two distinct integers and $$$c$$$ is a string ($$$1 \le i, j \le n$$$; $$$i \neq j$$$; $$$c$$$ is either imposter or crewmate).

There can be multiple comments for the same pair of $$$(i, j)$$$.

It is guaranteed that the sum of all $$$n$$$ does not exceed $$$2 \cdot 10^5$$$ and the sum of all $$$m$$$ does not exceed $$$5 \cdot 10^5$$$.

Output format:
For each test case, print one integer — the maximum possible number of imposters. If the comments contradict each other, print $$$-1$$$.

Constraints:
Time limit: 3.0 s
Memory limit: 256.0 MB

Note:
In the first test case, imposters can be Andreas $$$2$$$ and $$$3$$$.

In the second test case, imposters can be Andreas $$$1$$$, $$$2$$$, $$$3$$$ and $$$5$$$.

In the third test case, comments contradict each other. This is because player $$$1$$$ says that player $$$2$$$ is an imposter, and player $$$2$$$ says that player $$$1$$$ is a crewmate. If player $$$1$$$ is a crewmate, then he must be telling the truth, so player $$$2$$$ must be an imposter. But if player $$$2$$$ is an imposter then he must be lying, so player $$$1$$$ can't be a crewmate. Contradiction.

Example input:
5
3 2
1 2 imposter
2 3 crewmate
5 4
1 3 crewmate
2 5 crewmate
2 4 imposter
3 4 imposter
2 2
1 2 imposter
2 1 crewmate
3 5
1 2 imposter
1 2 imposter
3 2 crewmate
3 2 crewmate
1 3 imposter
5 0

Example output:
2
4
-1
2
5

Editorial:
If person $$$A$$$ said in a comment that person $$$B$$$ is a $$$crewmate$$$ then $$$A$$$ and $$$B$$$ belong to the same team (either imposters or crewmates).
If person $$$A$$$ said in a comment that person $$$B$$$ is an $$$imposter$$$ then $$$A$$$ and $$$B$$$ belong to different teams.
Solution $$$1$$$:
You can build a graph and check if all its components are bipartite.
If person $$$A$$$ said that $$$B$$$ is an imposter then we add an edge from $$$A$$$ to $$$B$$$.
If person $$$A$$$ said that $$$B$$$ is a crewmate then we add an edge from $$$A$$$ to a fake node and from the same fake node to $$$B$$$.
For each component, we check if it's bipartite and take the maximum from the two colours.
If a component is not bipartite then the answer is $$$-1$$$.
Solution $$$2$$$:
We can build the graph in the other way:
If $$$A$$$ and $$$B$$$ are in the same team then we add edge with weight $$$0$$$, otherwise with weight $$$1$$$.
Then you can use dfs and colour the nodes either $$$0$$$ or $$$1$$$ maintaining the property that $$$u \oplus v = w(u, v)$$$.