Description
In this assignment you will explore clustering: hierarchical and point-assignment. You will also experiment
with high dimensional data.
You will use three data sets for this assignment:
• https://www.cs.utah.edu/˜jeffp/teaching/cs5140/A3/C1.txt
• https://www.cs.utah.edu/˜jeffp/teaching/cs5140/A3/C2.txt
• https://www.cs.utah.edu/˜jeffp/teaching/cs5140/A3/C3.txt
These data sets all have the following format. Each line is a data point. The lines have either 3 or 6 tab
separated items. The first one is an integer describing the index of the points. The next 2 (or 5 for C3) are
the coordinates of the data point. C1 and C2 are in 2 dimensions, and C3 is in 5 dimensions. C1 should have
n=20 points, C2 should have n=1004 points, and C3 should have n=1000 points. We will always measure
distance with Euclidean distance.
As usual, it is highly recommended that you use LaTeX for this assignment. If you do not, you may
lose points if your assignment is difficult to read or hard to follow. Find a sample form in this directory:
https://www.cs.utah.edu/˜jeffp/teaching/latex/
1 Hierarchical Clustering (25 points)
There are many variants of hierarchical clustering; here we explore 3. The key difference is how you measure
the distance d(S1, S2) between two clusters S1 and S2.
Single-Link: measures the shortest link d(S1, S2) = min
(s1,s2)∈S1×S2
ks1 − s2k2.
Complete-Link: measures the longest link d(S1, S2) = max
(s1,s2)∈S1×S2
ks1 − s2k2.
Mean-Link: measures the distances to the means. First compute a1 =
1
|S1|
P
s∈S1
s and a2 =
1
|S2|
P
s∈S2
s then
d(S1, S2) = ka1 − a2k2 .
A (25 points): Run all hierarchical clustering variants on data set C1.txt until there are k = 4 clusters,
and report the results as sets.
Which variant did the best job, and which was the easiest to compute (think if the data was much larger)?
Explain your answers.
2 Point Assignment Clustering (50 points)
Point assignment clustering works by assigning every point x ∈ X to the closest cluster centers C. Let
φC : X → C be this assignment map so that φC(x) = arg minc∈C d(x, c). All points that map to the same
cluster center are in the same cluster.
Two good heuristics for these types of cluster are the Gonzalez (Algorithm 9.4.1) and k-Means++
(Algorithm 10.1.2) algorithms.
A: (20 points) Run Gonzalez and k-Means++ on data set C2.txt for k = 3. To avoid too much
variation in the results, choose c1 as the point with index 1.
Report the centers and the subsets for Gonzalez. Report:
• the 3-center cost maxx∈X d(x, φC(x)) and
• the 3-means cost P
x∈X(d(x, φC(x)))2
For k-Means++, the algorithm is randomized, so you will need to report the variation in this algorithm.
Run it several trials (at least 20) and plot the cumulative density function of the 3-means cost. Also report
what fraction of the time the subsets are the same as the result from Gonzalez.
B: (20 points) Recall that Lloyd’s algorithm for k-means clustering starts with a set of k centers C and
runs as described in Algorithm 10.1.1.
• Run Lloyds Algorithm with C initially with points indexed {1,2,3}. Report the final subset and the
3-means cost.
• Run Lloyds Algorithm with C initially as the output of Gonzalez above. Report the final subset and
the 3-means cost.
• Run Lloyds Algorithm with C initially as the output of each run of k-Means++ above. Plot a cumulative density function of the 3-means cost. Also report the fraction of the trials that the subsets are
the same as the input.
C: (10 points) Consider a set of points S ⊂ R
d
and d the Euclidean distance. Prove that
arg min
p∈Rd
X
x∈S
(d(x, p))2 =
1
|S|
X
x∈S
x.
Here are some suggested steps to follow towards the proof (note there are also other valid ways to prove
this, but, for instance, achieving some of these steps will get you partial credit):
1. First prove the same results for S ∈ R
1
.
2. Expand each term (d(x, p))2 = (x − p)
2 = x
2 + p
2 − 2xp.
3. Add the above terms together and take the first derivative.
4. Show the results for each dimension can be solved independently (use properties of edge lengths in a
right triangle).
3 k-Median Clustering (25 points)
The k-median clustering problem on a data set P is to find a set of k-centers C = {c1, c2, . . . , ck} to
minimize Cost1(P, C) = P
p∈P d(p, φC(p)). We did not explicitly talk much about this formulation in
class, but the techniques to solve it are all typically extensions of approaches we did talk about. This
problem will be more open-ended, and will ask you to try various approaches to solve this problem. We will
use data set C3.txt.
A: (20 points) Find a set of 4 centers C = {c1, c2, c3, c4} for the 4-medians problem on dataset C3.txt.
Report the set of centers, as well as Cost1(P, C). The centers should be in the write-up you turn in, but also
in a file formatted the same was as the input so we can verify the cost you found. That is each line has 1
center with 6 tab separated numbers. The first being the index (e.g., 1, 2, 3 or 4), and the next 5 being the
5-dimensional coordinates of that center.
Your score will be based on how small a Cost1(P, C) you can find. You can get 15 points for reasonable
solution. The smallest found score in the class will get all 20 points. Other scores will obtain points in
between.
Very briefly describe how you found the centers.
B: (5 points) Run your algorithm again for the 5-medians problem on dataset C3.txt. Report the set of
5 centers and the Cost1(P, C). You do not need to turn in a file for these, just write it in your report.
4 BONUS (2 points)
Recall that the k-center problem is to find a set of k centers C to minimize
Cost0(P, C) = max
p∈P
min
c∈C
d(p, c).
Let C
∗ be the optimal choice of k centers for the k-center problem, and let V
∗ = Cost0(P, C∗
).
Prove that the Gonzalez algorithm always finds a set of k centers C such that
Cost0(P, C) ≤ 2V
∗
