Assignment 5: Greedy Algorithms

Learning Goals

  • Implement an algorithm for solving the minimum spanning tree problem.
  • Learn how to analyze and develop greedy algorithms.

Grading

This assignment will be graded as pass/no-pass by a TA. To pass the assignment, you must

  1. Complete every question in the PDF and submit your answers to gradescope.
  2. Complete the programming component and submit your answers to gradescope.
  3. Meet with your TA during your learning community sessions. I prefer both partners to be present during the walk-through, but you can each meet with the TA separately if needed.
  4. Walk the TA through your answers. Do not expect to make corrections during the walk-through.
  5. The TA will then either
    • mark your assignment as 100% on gradescope, or
    • inform you that you have some corrections to make.
  6. If corrections are needed, then you will need to complete them and conduct a new walk-through with a TA.

If you have concerns about the grading walk-through, you can meet with me after you’ve first met with a TA.

Overview

This assignment includes two parts:

  1. Answer the question in this PDF.
  2. Write a program for the problem described in section Minimum Spanning Trees (must be compatible with Python 3.8).

You may work on this assignment alone or with a partner (I much prefer you to work with a partner). Please send me a message on Slack if you want me to assign you to work with a random partner. Whether you work alone or with a partner, only one partner will submit a both parts to gradescope.

Three key points:

  1. you should either print the PDF and write your answers directly on the printed sheet, or add annotations to the PDF with, for example, Edge (Windows) or Preview (OSX) (using the correct format and spacing will help us grade),
  2. ensure that the uploading partner adds the other partners after submission, and
  3. if you have any questions please ask them on Slack.

Example Execution

I always like to start by showing you the end result.

If you do not see an animation above this line (or if you see the animation but you don’t see the progress bar), you will need to refresh the page (sometimes more than once). Or you can go directly to the player: https://asciinema.org/a/367833

You can copy and past from the above animation!

Minimum Spanning Trees

Your program will accept two program arguments: (1) a filename and (2) a start vertex number. If you don’t need it, your program can ignore the start vertex (i.e., if you are implementing Kruska’ls algorithm). Your program will output the cost of the minimum spanning tree. You should name your file mst.py.

The input file

Your programs will read a file the file given by a user. The file will include all of the information needed to create an undirected graph with edge costs. The file has the following format:

[n] [m]
[v_i] [v_j] [c1]
[v_a] [v_b] [c2]

Where the first line gives values for n and m, which are the number of vertices and number of edges, respectively. Each line after the first gives two vertex numbers (v_i and v_j; v_a and v_b) and then a cost for that edge (cx). Here is an example input file (it is also attached):

5 7
1 2 2
1 4 6
2 3 3
2 4 8
2 5 5
3 5 7
4 5 9

Example Graph from Input File

Here are my test files.

Your program

You may implement a \(O(m lg n)\) version of Prim’s algorithm a \(O(m lg m)\) version of Kruskal’s algorithm or one of the \(O(mn)\) versions discussed in class. Your grade will be the same either way. But I encourage you to think about the more efficient versions.

Submitting Your Assignment

You will submit your PDF assignment on gradescope. Only one partner should submit your PDF. The submitter will add the other partner through the gradescope interface.

To pass the autograder, your output must exactly match the expected output. Your program output should be similar to the example execution above, and the autograder on gradescope will show you the correct output if yours is not formatted properly. You can use text-compare to compare your output to the expected output and that should give you an idea if you have a misspelled word or extra space (or if I do).

Additional details for using gradescope can be found here: