2018/round2/ethan_path.html

<p>
Ethan is doing his third programming assignment: finding the shortest path between two nodes in a graph.
</p>

<p>
Given an undirected, weighted graph with <strong>N</strong> nodes (numbered from 1 to <strong>N</strong>), having no self-loops or duplicate edges, 
Ethan must compute the length of the shortest path from node 1 to node <strong>N</strong>. Ethan has implemented an algorithm to solve this problem, described by the following pseudocode:
</p>

<ol>
<li>Set <em>i</em> to be equal to 1, and <em>d</em> to be equal to 0</li>
<li>If <em>i</em> is equal to <strong>N</strong>, output <em>d</em> and stop</li>
<li>Find the edge incident to node <em>i</em> that has the smallest weight
(if no edges are incident to <em>i</em> or if there are multiple such edges tied with the smallest weight, then crash instead)</li>
<li>Increase <em>d</em> by the weight of this edge, and set <em>i</em> to be equal to the other node incident to this edge</li>
<li>Return to Step 2</li>
</ol>

<p>
Since you were nice to Ethan on his second assignment, and since that encouragement clearly hasn't helped improve the quality of his code, you'd like to find a graph that shows as clearly as
possible why this solution is incorrect.
</p>

<p>
You're given the number of nodes in the graph <strong>N</strong>, as well as the maximum allowable edge weight <strong>K</strong> 
(each edge's weight must be an integer in the interval [1, <strong>K</strong>]). 
Under these constraints you want to maximize the absolute difference between Ethan's output and the actual shortest distance between nodes 1 and <strong>N</strong>. 
However, you don't want Ethan's algorithm to either crash or run forever. 
Note that node <strong>N</strong> must actually be reachable from node 1 in the graph, though the graph may be otherwise disconnected.
You can output any valid graph which gets the job done.
</p>


<h3>Input</h3>

<p>
Input begins with an integer <strong>T</strong>, the number of graphs.
For each graph, there is a line containing the space-separated integers <strong>N</strong> and <strong>K</strong>.
</p>


<h3>Output</h3>

<p>
For the <em>i</em>th graph, first output a line containing "Case #<em>i</em>: " 
followed by the maximum possible absolute difference between Ethan's algorithm's output and the correct answer.
Then, output a line containing as single integer <strong>E</strong>, the number of edges in your chosen graph which yields the above maximum absolute difference.
Then, output <strong>E</strong> lines, the <em>j</em>th of which contains three integers
<strong>U<sub>j</sub></strong>, <strong>V<sub>j</sub></strong>, and <strong>W<sub>j</sub></strong> denoting that
there is an edge between nodes <strong>U<sub>j</sub></strong> and <strong>V<sub>j</sub></strong> with weight <strong>W<sub>j</sub></strong>.
</p>

<p>
Note that there must be no self-loops (no edge may connect a node to itself), and no two edges may connect the same unordered pair of nodes.
</p>


<h3>Constraints</h3>

<p>
1 &le; <strong>T</strong> &le; 200 <br />
2 &le; <strong>N</strong> &le; 50 <br />
1 &le; <strong>K</strong> &le; 50 <br />
</p>


<h3>Explanation of Sample</h3>

<p>
In the first case, there are exactly two possible valid graphs, either of which would be accepted:
</p>

<pre>
  1
  1 2 1
  
  1
  2 1 1
</pre>
  
<p>
In each of the above graphs, Ethan's algorithm's answer and the correct answer are both equal to 1. 
There's an absolute difference of 0 between those answers, which is the maximum possible absolute difference.
</p>

<p>
In the second case, one possible graph which would be accepted is as follows (with Ethan's algorithm's answer and the correct answer both equal to 42):
</p>

<pre>
  1
  1 2 42
</pre>

<p>
In the third case, one possible graph which would be accepted is as follows (with Ethan's algorithm's answer and the correct answer both equal to 1):
</p>

<pre>
  3
  1 2 2
  4 1 1
  4 2 1
</pre>
  
<p>
Putting those together, the following is one possible sequence of outputs for the first 3 cases which would be accepted:
</p>

<pre>
  Case #1: 0
  1
  1 2 1
  Case #2: 0
  1
  1 2 42
  Case #3: 0
  3
  1 2 2
  4 1 1
  4 2 1
</pre>

<p>
Do not output the line "<strong>Multiple possible accepted graphs</strong>".
</p>