2019/quals/service_trees.html

<p>
Carlos is hoping to strike it rich with his new consulting firm, Carlos 
Structures Industries (CSI). With a solid mission statement and a can-do
attitude, he's sure to succeed in the wide world of bespoke data structures.
</p>

<p><em>
"Our vision is to synergistically leverage our core competencies to deliver
competitive market solutions that assertively meet our clients' needs."
</em></p>

<p>
His first client, the well-known translation firm Treehouse, is looking to
update their logo. Treehouse wants their logo to be a rooted tree with
<strong>N</strong> nodes numbered 1 through <strong>N</strong>. They have a rough idea of what the tree should
look like, but they want Carlos to finish fleshing it out. In particular,
they have <strong>M</strong> requirements, the <em>i</em>th of which states that the
<a href="https://en.wikipedia.org/wiki/Lowest_common_ancestor" target="_blank">lowest common ancestor</a>
of nodes <strong>X<sub>i</sub></strong> and <strong>Y<sub>i</sub></strong> must be node <strong>Z<sub>i</sub></strong>.
</p>

<p>
Carlos's goal is to find any valid tree consistent with all of these requirements
if possible, or to determine that no such tree exists.
</p>


<h3>Input</h3>

<p>
Input begins with an integer <strong>T</strong>, the number of tree designs.
For each design, there is first a line containing the space-separated integers <strong>N</strong> and <strong>M</strong>. 
Then <strong>M</strong> lines follow, the <em>i</em>th of which contains the space-separated integers
<strong>X<sub>i</sub></strong>, <strong>Y<sub>i</sub></strong>, and <strong>Z<sub>i</sub></strong>.

</p>


<h3>Output</h3>

<p>
For the <em>i</em>th design, print a line containing "Case #<em>i</em>: "
followed by a description of a valid rooted tree if possible, or the string "Impossible" if no valid tree exists. 
A tree description is <strong>N</strong> space-separated integers, the <em>j</em>th of which is node <em>j</em>'s parent (or 0 if node <em>j</em> is the root).
</p>


<h3>Constraints</h3>

<p>
1 &le; <strong>T</strong> &le; 100 <br />
2 &le; <strong>N</strong> &le; 60 <br />
1 &le; <strong>M</strong> &le; 120 <br />
1 &le; <strong>X<sub>i</sub></strong>, <strong>Y<sub>i</sub></strong>, <strong>Z<sub>i</sub></strong> &le; <strong>N</strong> <br />
<strong>X<sub>i</sub></strong> &ne; <strong>Y<sub>i</sub></strong>
</p>

<h3>Explanation of Sample</h3>

<p>
In the first case, the LCA of nodes 1 and 2 in the chosen tree is 3, as required. This is the only valid output for this case.
</p>

<p>
In the third case, "2 3 0 3" would also be accepted, while no other outputs would be.
</p>

<p>
In the fifth and sixth cases, multiple possible outputs would be accepted.
</p>