| <p> |
| With a successful commission under his belt, Carlos has really made a splash |
| with his consulting firm, Carlos Structures Industries. His next customer is |
| Texas Instruments, the well-known manufacturer of graphing calculators. |
| </p> |
|
|
| <p> |
| <em> |
| "In today's competitive environment, you need an edge. I can make sure that |
| your newest graphing calculator is chock-full of the best modern graphs." |
| </em> |
| </p> |
|
|
| <p> |
| Carlos's pitch seems to have worked as Texas Instruments has ordered an |
| undirected, weighted graph. Their R&D department has come up with a list |
| of requirements that will ensure the graph is a hit with Gen Z schoolchildren. |
| </p> |
|
|
| <p> |
| To start with, the graph must have <strong>N</strong> nodes numbered 1 to <strong>N</strong>. It must have no |
| self-loops and at most one edge connecting each unordered pair of nodes. The |
| weight of each edge must be an integer between 1 and 1,000,000, inclusive. |
| The graph does not need to be connected. |
| </p> |
|
|
| <p> |
| The graph must also satisfy <strong>M</strong> customer requirements, the <em>i</em>th of which states |
| that the shortest distance between two different nodes <strong>X<sub>i</sub></strong> and <strong>Y<sub>i</sub></strong> must be equal to <strong>Z<sub>i</sub></strong>. |
| No two requirements pertain to the same unordered pair of nodes. |
| </p> |
|
|
| <p> |
| Carlos's goal is to find any valid graph consistent with all of these requirements |
| if possible, or to determine that no such graph exists. |
| </p> |
|
|
|
|
|
|
| <h3>Input</h3> |
|
|
| <p> |
| Input begins with an integer <strong>T</strong>, the number of graphs that Texas Instruments has commissioned. |
| 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 <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 graph, print a line containing "Case #<em>i</em>: " |
| followed by either an integer <strong>E</strong> and then a description of a valid graph if possible, or the string "Impossible" if no valid graph exists. |
| </p> |
|
|
| <p> |
| A graph description contains <strong>E</strong> lines, where <strong>E</strong> is the number of edges in your graph. |
| The <em>i</em>th line contains the space-separated integers |
| <strong>A<sub>i</sub></strong>, <strong>B<sub>i</sub></strong>, and <strong>W<sub>i</sub></strong> |
| indicating that there is an edge between nodes <strong>A<sub>i</sub></strong> and <strong>B<sub>i</sub></strong> with weight <strong>W<sub>i</sub></strong>. |
| Please keep in mind that your graph must satisfy all of the requirements stated above (both the fundamental requirements dictated by Texas Instruments, and the <strong>M</strong> customer ones). |
| </p> |
|
|
|
|
| <h3>Constraints</h3> |
|
|
| <p> |
| 1 ≤ <strong>T</strong> ≤ 350 <br /> |
| 2 ≤ <strong>N</strong> ≤ 50 <br /> |
| 1 ≤ <strong>M</strong> ≤ 1,000 <br /> |
| 1 ≤ <strong>X<sub>i</sub></strong>, <strong>Y<sub>i</sub></strong> ≤ <strong>N</strong> <br /> |
| <strong>X<sub>i</sub></strong> ≠ <strong>Y<sub>i</sub></strong> <br /> |
| 1 ≤ <strong>Z<sub>i</sub></strong> ≤ 1,000,000 <br /> |
| </p> |
|
|
| <h3>Explanation of Sample</h3> |
|
|
| <p> |
| In the graph described by the first sample case's sample output, the shortest distance between nodes 3 and 1 is 5 (along the path 3 -> 2 -> 1), as required. <b>Multiple other outputs would also be accepted for this case.</b> |
| </p> |
|
|
| <p> |
| <b>Multiple other outputs would also be accepted for the third and fourth cases.</b> |
| </p> |
|
|
|
|
