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Overview:

The purpose of this assignment is to experience some of the problems involved with implementing an algorithm (in this case, a minimum spanning tree algorithm) in practice. As an added benefit,
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 each submit a copy of the report in Gradescope.)
I recommend using a common programming language such as Java, C, or C++. If you use a more
obscure language, that is fine, but if there errors that are correspondingly harder to find it may cause your
grade to be lower.

I might advise you not to use a scripting language like Python, although some students
have successfully done the assignment in Python.
We will be considering complete, undirected graphs. A graph with n vertices is complete if all possible
n
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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 first goal is to determine in each case how the expected (average) weight of the minimum
spanning tree (not an edge, the whole MST) 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, say with a value from the machine’s clock.) 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) = log n, f(n) = 1.5

n, or f(n) = 2n
log n
. 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 accepts the following
command line form:
./randmst 0 numpoints numtrials dimension
The flag 0 is meant to provide you some flexibility; you may use other values for your own testing,
debugging, or extensions. Note: you should test your program – design tests to make sure your program

is working, for example by checking it on some small examples (or find other tests of your choosing). You
don’t need to put your tests in your writeup, that is for you.

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 3/4 for cube and
hypercube; 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. Hence we require:
• If possible, for compatibility reasons, the code should run on the Harvard system (nice.fas.harvard.edu
cluster of Linux machines – you can use an ssh client to log in if you have a Harvard account), even
if you code on another system.

• We expect an executable with the code as described above (input as described, answers should go to
standard output as described).
• The code should compile with make; no instructions for humans. That is, the command “make
randmst” should produce an executable from your directory. You may need to read up on makefiles
to make this happen.
What to hand in: Besides submitting a copy of the code you created, your group should hand in a
single well organized and clearly written report describing your results. For the first part of the assignment,
this report must contain the following quantitative results (for each graph type):

• A table listing the average tree size for several values of n. (A graph is insufficient, although you
can have that too; we need to see the actual numbers. If you have a graph but no table of actual
numbers, I will take points off.)
• A description of your guess for the function f(n).
Run your program for n = 128; 256; 512; 1024; 2048; 4096; 8192; 16384; 32768; 65536; 131072; 262144;
and larger values, if your program runs fast enough. (Having your code handle up to at least n = 262144
vertices is one of the assignment requirements; however, handling n up to 131072 will result in only losing
1-2 points.

Providing results only for smaller n will hurt your score on the assignment.) Run each value
of n at least five times and take the average. (Make sure you re-seed the random number generator
appropriately!)
In addition, you are expected to discuss your experiments in more depth. This discussion should
reflect 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?
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• 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 effect?
• 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!
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