Overview: The purpose of this assignment is to experience some of the problems involved with imple-
menting an algorithm (in this case, a minimum spanning tree algorithm) in practice. As an added bene t,
we will explore how minimum spanning trees behave in random graphs.
Assignment: You may work in groups of two, or by yourself. Both partners will receive the same
grade and turn in a single joint report. You should program in Python 3.
We will be considering complete, undirected graphs. A graph with n vertices is complete if all possible
n
2
edges are present in the graph.
Consider the following types of graphs:
Complete graphs on n vertices, where the weight of each edge is a real number chosen uniformly at
random on [0;1].
Complete graphs on n vertices, where the vertices are points chosen uniformly at random inside the
unit square. (That is, the points are (x;y), with x and y each a real number chosen uniformly at
random from [0;1].) The weight of an edge is just the Euclidean distance between its endpoints.
Complete graphs on n vertices, where the vertices are points chosen uniformly at random inside the
unit cube (3 dimensions) and hypercube (4 dimensions). As with the unit square case above, the
weight of an edge is just the Euclidean distance between its endpoints.
Your rst goal is to determine in each case how the expected (average) weight of the minimum spanning
tree grows as a function of n. This will require implementing an MST algorithm, as well as procedures that
generate the appropriate random graphs. (You should check to see what sorts of random number generators
are available on your system, and determine how to seed them appropriately.) You may implement any
MST algorithm (or algorithms!) you wish; however, I suggest you choose carefully.
For each type of graph, you must choose several values of n to test. For each value of n, you must run
your code on several randomly chosen instances of the same size n, and compute the average value for your
runs. Plot your values vs. n, and interpret your results by giving a simple function f(n) that describes
your plot. For example, your answer might be f(n) = logn, f(n) = 1:5pn, or f(n) = 2nlogn. Try to make
your answer as accurate as possible; this includes determining the constant factors as well as you can. On
the other hand, please try to make sure your answer seems reasonable.
Code setup:
So that we may test your code ourselves as necessary, please make sure your code can be run from the
command line via the following command:
python randmst.py 0 numpoints numtrials dimension
The ag 0 is meant to provide you some exibility; you may use other values for your own testing,
debugging, or extensions. The value numpoints is n, the number of points; the value numtrials is the
number of runs to be done; the value dimension gives the dimension. (Use dimension = 2 for the square,
and 3 or 4 for cube and hypercube respectively; use dimension = 0 for the case where weights are assigned
randomly. Notice that dimension 1 is just not that interesting.) The output for the above command line
should be the following:
average numpoints numtrials dimension
where average is the average minimum spanning tree weight over the trials.
Please pay attention to the following requirements. In order to grade appropriately, our objective is
to ensure that we can run the programs without any special per-student attention.
What to hand in: You (our your group, if you work with a partner) should submit a zip le
through Canvas containing your python program randmst.py, and a single pdf containing a well organized
and clearly written report describing your results. If you work with a partner, do not forget to include
both of your names at the top of the pdf.
The report must consist of two parts. The rst part of the report must contain the following quanti-
tative results (for each graph type):
A table or graph listing the average tree size for several values of n.
A description of your guess for the function f(n).
Run your program for n = 16; 32; 64; 128; 256; 512; 1024; 2048; 4096; 8192; 16384 and larger values,
if your program runs fast enough. Run each value of n at least ve times and take the average. (Make
sure you re-seed the random number generator appropriately, so that you don’t get the same \random"
numbers out of the generator on di erent runs).
For the second part of the report, you are expected to discuss your experiments in more depth. This
discussion should re ect what you have learned from this assignment; the actual issues you choose to
discuss are up to you. Here are some possible suggestions for the second part:
Which algorithm did you use, and why?
Are the growth rates (the f(n)) surprising? Can you come up with an explanation for them?
How long does it take your algorithm to run? Does this make sense? Do you notice things like the
cache size of your computer having an e ect?
Did you have any interesting experiences with the random number generator? Do you trust it?
Your grade will be based primarily on the correctness of your program and your discussion of the
experiments. Other considerations will include the size of n your program can handle. Please do a careful
job of solid writing in your writeup. Length will not earn you a higher grade, but clear descriptions of
what you did, why you did it, and what you learned by doing it will go far.
Hints:
To handle large n, you may want to consider simplifying the graph. For example, for the graphs in
this assignment, the minimum spanning tree is extremely unlikely to use any edge of weight greater than
k(n), for some function k(n). We can estimate k(n) using small values of n, and then try to throw away
edges of weight larger than k(n) as we increase the input size. Notice that throwing away too many edges
may cause problems. Why will throwing away edges in this manner never lead to a situation where the
program returns the wrong tree?
You may invent any other techniques you like, as long as they give the same results as a non-optimized
program. Be sure to explain any techniques you use as part of your discussion and attempt to justify why
they should give the same results as a non-optimized program!
Library. In particular, you may not use packages such as NetworkX. There are three reasons for this.
First, the point of the assignment is to make you do most of the work of e ciently implementing an MST
algorithm from scratch, and you should be able to do this without using any package downloaded from
online. Second, most of these libraries have a lot of memory overhead, which are likely to cause you trouble
as you try to scale to large graphs. That will make it hard or impossible to get full credit on the problem
set. Finally, as a logistical issue, I do not want to have to ask the graders to install external libraries in
order to grade the problem set.