[
{('Julia', 'Spike Spiegel'), ('Faye Valentine', 'Edward')},
{('Spike Spiegel', 'Vicious') },
{('Jet Black', 'Ein'), ('Edward', 'Julia')},
{('Vicious', 'Jet Black')}
])

```
The previous question only dealt with which games can be played at the same time, but not on the
order of the game groups. The preference is that if a player is also a referee then they play all of their
games before refereeing. Your task is to determine if it is possible to order the game groups so that
this is true.
```

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Answer from Sia

Posted 6 months ago

Solution by Steps

step 1

To determine if it is possible to order the game groups so that a player who is also a referee plays all their games before refereeing, we need to model this as a directed graph problem

step 2

Represent each game as a vertex in a directed graph. Draw a directed edge from vertex $A$ to vertex $B$ if a player in game $A$ is also a referee in game $B$

step 3

The goal is to check if this directed graph has a topological ordering. A topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge $uv$ from vertex $u$ to vertex $v$, $u$ comes before $v$ in the ordering

step 4

To find a topological ordering, we can use Kahn's algorithm or Depth-First Search (DFS). If the graph has a cycle, then a topological ordering is not possible, and thus it is not possible to order the game groups as required

step 5

Apply Kahn's algorithm:
1. Compute the in-degree of each vertex.
2. Collect all vertices with in-degree 0 into a queue.
3. While the queue is not empty:
a. Remove a vertex from the queue and add it to the topological order.
b. For each outgoing edge from this vertex, reduce the in-degree of the destination vertex by 1.
c. If the in-degree of the destination vertex becomes 0, add it to the queue.
4. If all vertices are processed, a topological order exists; otherwise, a cycle is present

step 6

If a topological order exists, it is possible to order the game groups so that a player who is also a referee plays all their games before refereeing. If no topological order exists, it is not possible

Answer

It is possible to order the game groups so that a player who is also a referee plays all their games before refereeing if and only if the directed graph representing the games and their dependencies has a topological ordering.

Key Concept

Topological Ordering in Directed Graphs

Explanation

A topological ordering of a directed graph ensures that for every directed edge $uv$, vertex $u$ comes before vertex $v$. This property is crucial for scheduling tasks with dependencies, such as ensuring a player plays all their games before refereeing.

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