# Lab 12: Railroad Construction

## Specification

There are _n_ cities that want to create a shared train network so that each
pair of cities is connected by rail to each of the other cities. The cost of
laying track is directly proportional to the distance between two cities. The
cost to lay one mile of railroad track is $1M. Your task is to determine the
lowest cost possible for laying the track so that all the cities are connected.

## Input

A single line containing the number n of cities followed by n city coordinate
lines. Each of the coordinate lines is a pair of floating point numbers x y
giving the coordinates of the cities on a square grid (for this problem you may
assume the earth is flat and that the units of the grid structure are given in
miles).

## Output

The minimum amount of money (in millions) that must be spent to create the
railroad network. You can round this to a single decimal place.

<table>
    <tr>
        <td>Sample Input</td>
        <td>Sample Output</td>
    </tr>
    <tr>
        <td><pre>
3
0 1.5
1 0
0 0</pre></td>
        <td><pre>$2.5M</pre></td>
    </tr>
    <tr>
        <td><pre>
8
0 0
1 1
0 1
3 3
4 5
2 2
1 0
3 2</pre></td>
        <td><pre>$8.7M</pre></td>
    </tr>
</table>

## Hints

1. You may use our [connected_components.py](./connected_components.py) Download
   connected_components.py in your code if you want., which will produce labels
   for each of the vertices that correspond to their component. You will need to
   use the adjacency list structure as suggested below if you wish to use this
   code in its unmodified state.
2. You may have already envisioned a way to represent this problem as a graph.
   Draw the graph in the small example above on paper. How many edges does it
   have? If you are unsure of this, check with me before you start coding the
   solution.

## Adjacency List

Augment your dictionary/set based adjacency structure to use a dictionary for
the edges (instead of a set as before). This will allow you to store the edge
weights as the values of this second dictionary.

## Method/Programming requirement

Your program must implement either Borůvka's algorithm (I suggest you use
Borůvka's) or Prims (not Kruskal). You must specify which algorithm you are
implementing in the comments. Any other method to compute the answer will result
in zero credit.

## Submission

Submit the code in a file named [cs412_railroad_a.py](./cs412_railroad_a.py) to
Gradescope. If you use the supplied copy of
[connected_components.py](./connected_components.py), you only need to import it
into your code (it will be available on Gradescope, so, no need to submit). If
you change connected_components.py, then please rename it and submit it as part
of your code (or you can copy/paste the code into your lab and just submit a
single file, it is up to you). You can import the code using the following
python:

```
from connected_components import count_and_label
```