COMP9318 (20T1) ASSIGNMENT 1 Solved

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Q1. (40 marks)

Consider the following base cuboid Sales with four tuples and the aggregate function

SUM:

Location, Time, and Item are dimensions and Quantity is the measure. Suppose the system has built-in support for the value ALL.

  1. (1)  List the tuples in the complete data cube of R in a tabular form with 4 attributes, i.e., Location, T ime, Item, SUM(Quantity)?
  2. (2)  Write down an equivalent SQL statement that computes the same result (i.e., the cube). You can only use standard SQL constructs, i.e., no CUBE BY clause.
  3. (3)  Consider the following ice-berg cube query:
              SELECT  Location, Time, Item, SUM(Quantity)
    
          FROM    Sales
          CUBE BY Location, Time, Item
          HAVING  COUNT(*) > 1

Draw the result of the query in a tabular form.
(4) Assume that we adopt a MOLAP architecture to store the full data cube of R, with

the following mapping functions:
1 if x = ‘Sydney’,

2 if x = ‘Melbourne’, 0 ifx=ALL.

1 if x = 2005, 

Location

T ime

I tem

Quantity

Sydney Sydney Sydney Melbourne

2005
2006
2006
2005

PS2 PS2 Wii XBox 360

1400
1500
 500
1700

fLocation(x) =

fTime(x) = 2 if x = 2006,

0 ifx=ALL. 1

2

DUE ON 20:59 12 APR, 2020 (SUN)

1 if x = ‘PS2’,
2 if x = ‘XBox 360’, 3 if x = ‘Wii’,
0 ifx=ALL.

Draw the MOLAP cube (i.e., sparse multi-dimensional array) in a tabular form of (ArrayIndex,Value). You also need to write down the function you chose to map a multi-dimensional point to a one-dimensioinal point.

Q2. (30 marks)

fItem(x) =

Consider the given similarity matrix. You are asked to perform group average hierar- chical clustering on this dataset.

You need to show the steps and final result of the clustering algorithm. You will show the final results by drawing a dendrogram. The dendrogram should clearly show the order in which the points are merged.

p1

p2

p3

p4

p5

p1 p2 p3 p4 p5

1.00 0.10 0.41 0.55 0.35

0.10 1.00 0.64 0.47 0.98

0.41 0.64 1.00 0.44 0.85

0.55 0.47 0.44 1.00 0.76

0.35 0.98 0.85 0.76 1.00

Q3. (30 marks)

1 2 3 4 5 6 7

8 9

10

Data: D is a dataset of n d-dimensional points; k is the number of clusters. Initialize k centers C = [c1, c2, . . . , ck];
canStop ← false;
while canStop = false do

Initialize k empty clusters G = [g1, g2, . . . , gk]; for each data point p ∈ D do

cx ← NearestCenter(p, C); gcx .append(p);

for each group g ∈ G do
ci ← ComputeCenter(g);

return G;

Algorithm 1: k-means(D, k)

Consider the (slightly incomplete) k-means clustering algorithm as depicted in Algo- rithm 1.

COMP9318 (20T1) ASSIGNMENT 1 3

(1) Assume that the stopping criterion is till the algorithm converges to the final k clusters. Can you insert several lines of pseudo-code to the algorithm to implement this logic? You are not allowed to change the first 7 lines though.

(2) The cost of k clusters is just the total cost of each group gi, or formally k

cost(g1, g2, . . . , gk) = 􏰁 cost(gi) i=1

cost(gi) is the sum of squared distances of all its constituent points to the center ci, or

cost(gi) = 􏰁 dist2(p, ci) p∈gi

dist() is the Euclidean distance. Now show that the cost of k clusters as evaluated at the end of each iteration (i.e., after Line 9 in the current algorithm) never increases. (You may assume d = 2)

(3) Prove that the cost of clusters obtained by k-means algorithm always converges to a local minima. You can make use of the previous conclusion even if you have not proved it.

Submission

Please write down your answers in a file named ass1.pdf. You must write down your name and student ID on the first page. You should typeset your answers in LATEX or MS Word. We do not accept handwritten answers.

You can submit your file by

  give cs9318 ass1 ass1.pdf

Hint 1. In fact, the two loops (Lines 5–7 and Lines 8–9) never increases the cost.

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