2016/finals/grundy_graph.html

<p>
Alice and Bob are spending the day in the local library, learning about 2-player zero-sum games. One of the books they're reading, "Grundy Numbers For Fun And Profit" by Nim Nimberly, 
has an interactive insert with a bunch of graphs and instructions for a game where the players take turns colouring each graph's vertices.
</p>

<p>
Each game starts with a directed graph that has 2*<strong>N</strong> vertices, numbered from 1 to 2*<strong>N</strong>, all of which are initially uncoloured, and <strong>M</strong> edges. 
The <strong>i</strong>th edge goes from vertex <strong>A<sub>i</sub></strong> to vertex <strong>B<sub>i</sub></strong>. No two edges connect the same pair of vertices
in the same direction, and no edge connects a vertex to itself.
</p>

<p>
Alice goes first and colours vertices 1 and 2. She must colour one of these two vertices black, and the other one white. Bob then takes his turn and similarly colours vertices 3 and 4, one of them black and the other one white. 
This continues with Alice colouring vertices 5 and 6, Bob colouring 7 and 8, and so on until every vertex is coloured.
At the end of the game, Alice wins if there are no edges going from a black vertex to a white one. Bob wins if such an edge exists.
</p>

<p>
Who will win if Alice and Bob play optimally?
</p>

<h3>Input</h3>
<p>
Input begins with an integer <strong>T</strong>, the number of graphs.
For each graph, there is first a line containing the space-separated integers <strong>N</strong> and <strong>M</strong>. 
Then <strong>M</strong> lines follow, the <strong>i</strong>th of which contains the space-separated integers
<strong>A<sub>i</sub></strong> and <strong>B<sub>i</sub></strong> .
</p>


<h3>Output</h3>
<p>
For the <strong>i</strong>th graph, print a line containing "Case #<strong>i</strong>: " followed by the winner of the game, either "Alice" or "Bob".
</p>


<h3>Constraints</h3>
<p>
1 &le; <strong>T</strong> &le; 45 <br />
1 &le; <strong>N</strong> &le; 500,000 <br />
0 &le; <strong>M</strong> &le; 500,000 <br />
1 &le; <strong>A<sub>i</sub></strong>, <strong>B<sub>i</sub></strong>, &le; 2*<strong>N</strong> <br />
</p>


<h3>Explanation of Sample</h3>
<p>
For the first graph, Alice can color vertex 1 white and vertex 2 black. Since all edges start at vertex 1, Alice will win.

For the second graph, Alice can't control the color of vertex 3. If Bob makes it white, then one of the two edges must be from a black vertex to a white vertex, so Bob wins.
</p>