omg hi Codeforces!

I've been meaning to write a blog about this interesting, but not (as far as I know) documented trick for a while. I've decided to call it the Amogus trick after the first problem where I encountered and used it.

Prerequisites

• DSU
• Basic knowledge of 2-SAT (definitely not required, but it may make the blog easier to understand)

Focus Problem: 1594D - The Number of Imposters

First, let's solve an easier version of this problem, where we just need to find whether there exists a configuration of player roles (i.e. Crewmate or Imposter) such that all the statements made by players so far are true.

Let's look at the two different types of statements made by players separately.

• Case 1: Player $$$i$$$ claims Player $$$j$$$ is a crewmate. Now, if Player $$$i$$$ is a crewmate, Player $$$j$$$ will also be a crewmate. Similarly, if Player $$$i$$$ is an imposter, Player $$$j$$$ will also be an imposter. The reversed versions of these statements are also true. Using this, we can create a virtual "edge" between Players $$$i$$$ and $$$j$$$, as their roles in the game will always be the same. More formally, for those familiar with 2-SAT or otherwise, we create the equivalency $$$v_i = v_j$$$.

• Case 2: Player $$$i$$$ claims Player $$$j$$$ is an imposter. Now, if Player $$$i$$$ is a crewmate, Player $$$j$$$ will be an imposter. Similarly, if Player $$$i$$$ is an imposter, Player $$$j$$$ will be a crewmate. We can create a virtual "anti-edge" between Players $$$i$$$ and $$$j$$$, as their roles in the game will always be the different. More formally, we create the equivalency $$$v_i = !v_j$$$.

Adding edges is easy, we can just use normal DSU. But how do we deal with anti-edges? This is where the Amogus Trick comes in!

We can deal with these anti-edges by creating a DSU with $$$2n$$$ nodes, where nodes $$$1$$$ to $$$n$$$ represent player $$$i$$$ being a crewmate, and nodes $$$n + 1$$$ to $$$2n$$$ represent player $$$i - n$$$ being an imposter. Now, let's look at those cases again.

• Case 1 results in both players having the same role in the game. Therefore, when such a statement is said, we can unite nodes $$$i$$$ and $$$j$$$, and similarly, unite nodes $$$i + n$$$ and $$$j + n$$$.

• Case 2 results in both players having differing roles in the game. Therefore, when such a statement is said, we can unite nodes $$$i$$$ and $$$j + n$$$, and similarly, unite nodes $$$i + n$$$ and $$$j$$$.

Testcases 2 & 3 of the sample input in the problem, respectively, visualised with the Amogus trick:

So, how can we solve our reduced problem with this? Note that a player can be exactly one of $$$\{ \text{Crewmate, Imposter} \}$$$, so a configuration is invalid iff for some $$$1 \le i \le n$$$, nodes $$$i$$$ and $$$i + n$$$ are in the same component in our DSU. This is the only condition we need to check, as since the edges we add to our DSU are symmetric, there will always be a valid assignment of roles.

It's not too difficult to extend this idea to solve our original problem.

Using this trick, we can solve a variety of other problems, such as dynamic bipartiteness checking, and it can often be paired with other modifications of DSU such as with support for rollbacks.

Other Problems:

Problems are ordered (roughly) in ascending order of difficulty.

1702E - Split Into Two Sets

1290C - Prefix Enlightenment

1713E - Cross Swapping

1444C - Team-Building

A special thank you to kostia244, BucketPotato, fishy15 and AlperenT for providing lots of feedback and/or sample problems!