Midterm CS345 Examination
Wednesday, May 14, 2003, 9:30 11:30AM
Directions
The exam is open book; any written materials may be used.
Answer all 9 questions on the exam paper itself.
The total number of points is 120 i.e., 1 point per minute.
Do not forget to sign the pledge below.
I acknowledge and accept the honor code.
Print your name here:
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Problem 1: 12 points Let
1 and 2 denote two columns items of a matrix that
represents market-basket data. Let 1 _ 2 denote a column that is the row-wise logical
OR of the two columns; i.e., 1 _ 2 has a 1 when either 1 or 2 or both has a 1, and has
0 otherwise. Similarly, 1 ^ 2 denotes the row-wise logical AND of the two columns; i.e.,
1 ^ 2 has a 1 if and only if both columns have 1. Let denote the minhash value for
column . That is, is the smallest such that the th row in the chosen permuted
order of rows has a 1 in column . For each of the following statements, indicate whether
it is always true or sometimes false, by circling T or F, respectively.
,
a 1 _ 2 = min 1 2 T F
,
b 1 ^ 2 = max 1 2 T F
c If 1 = 2, then 1 _ 2 = 1 T F
d If 1 = 2, then 1 ^ 2 = 1 T F
C
C
C
C
C
C
C
C
C
C
C
C
hC
C
hC
i
i
C
hC
C
hC
hC
C
hC
;h C
;h C
hC
hC
hC
C
hC
hC
hC
hC
C
hC
Problem 2: 12 points Let denote the set of pages with link to page , and let
In x
x
denote the set of pages to which page links. Let , , and denote the
hubbiness," authority, and PageRank of page , respectively. Indicate whether each of
the following statements is always true T or sometimes false F.
a If , then . T F
b If , then . T F
c If , then . T F
d If , then . T F
Out x
x
hx
ax
px
x
Out i
Out j
hi
hj
Out i
Out j
pi
pj
In i
In j
pi
pj
In j
In i
ai
aj
Problem 3: 15 points What are all the stable models for the following propositionallogic program?
p1
q1
p2
p2
q2
:::::-
NOT
NOT
p1
NOT
NOT
q1
p1
q2
p2
2
Problem 4: 15 points A collection of market-basket data has 100,000 frequent items,
and 1,000,000 infrequent items. Each pair of frequent items appears 100 times; each pair
consisting of one frequent and one infrequent item appears 10 times, and each pair of
infrequent items appears once. Answer each of the following questions. Your answers only
have to be correct to within 1, and for convenience, you may optionally use scienti c
notation, e.g., 3 14 108 instead of 314,000,000.
a What is the total number of pair occurrences? That is, what is the sum of the counts
of all pairs?
:
b We did not state the support threshold, but the given information lets us put bounds
on the support threshold . What are the tightest upper and lower bounds on ?
s
s
c Suppose we apply the PCY algorithm to this data. If the actual support threshold
is 10,000,000 i.e., 107, and pairs in each of the three categories distribute as evenly
as possible, what is the smallest number of buckets we can use so that most of the
buckets are not frequent?
s
Problem 5: 16 points Consider the following rules:
pX :- intX & X 2 & NOT cX
6= Y
Think of as meaning
is a prime" and as
is composite." The EDB
predicate says that is a positive integer, and in practice it will hold a nite set
of integers. The EDB predicate
means that evenly divides .
Suppose that
= f1 2 3 4g, and
is the expected relation on these four
integers; that is,
= f1 1 2 1 3 1 4 1 2 2 4 2 3 3 4 4g. If
we instantiate these rules in all possible ways, eliminate rules with a known false subgoal
and then eliminate known true subgoals from the remaining rules, we are left with the
following:
cX :- intX & pY & dividesX,Y & X
pX
X
int X
cX
X
X
divides X; Y
int
divides
;
;
;
;
Y
X
divides
;
;
;
;
p2 :- NOT c2
p3 :- NOT c3
p4 :- NOT c4
;
;
c2
c3
c4
c4
;
::::-
;
;
;
;
;
;
;
p1
p1
p1
p2
a the Use alternating- xedpoint method to compute the well-founded model for this
program plus EDB, by lling in the following table and then indicating the truth
value T, F, UNK of each of the eight ground atoms. The table may have extra space
for rounds that need not be computed; you may ll in the table only until you are
3
sure you have reached convergence.
0
1
Round
1
2
3
4
1
2
3
4
2
3
4
Truth Value
p
p
p
p
c
c
c
c
b In the space below, draw the dependency graph for the instantiated atoms and
for 1 4.
pi
ci
i
c Are the rules with the given EDB locally strati ed?
are; if not, describe an in nite negative path.
If so, tell what the strata
d Suppose contains the integers from 1 to , and
contains all those pairs
such that divides and and are integers between 1 and . For what values
of will the rules and EDB be locally strati ed? Explain brie y.
int
i; j
n
j
i
i
j
divides
n
n
4
Problem 6: 16 points A view-centric information system has a single view:
vX,Y,Z :- eX,Y & eY,Z & eX,Z
We wish to answer the following query:
qA,B,C,D :- eA,B & eB,C & eC,D & eA,C & eB,D & eA,D
Notice that in this unusual case, neither the view de nition nor the query have any variables
that do not appear in the head. That fact may simplify reasoning about the problem. Also
observe that the view describes a triangle in a graph, but the edges are directed, and go in
the direction from one argument of the head representing a node to another that appears
to the right, among the arguments of the head. Likewise, the query asks for a complete
graph of 4 nodes, again with direction determined by to the right, among the arguments
of the head."
A conjunctive query , all of whose subgoals have predicate , is a solution if, after
expansion, it is contained in the query. For to be a minimal solution, any conjunctive
query formed by deleting one or more subgoals from the body of must not be a
solution; i.e., the expansion of is not contained in the query. For each of the proposed
solutions below, tell whether it is: not a solution, a solution but not minimal, or
a minimal solution. In each case, explain your reasoning brie y. Suggestions: describe
the expansions of the proposed solutions and indicate containment mappings when needed.
a qA,B,C,D :- vA,B,C & vB,C,D
Q
v
Q
P
Q
P
i
ii
iii
b
qA,B,C,D :- vA,B,C & vB,C,D & vA,C,D
5
c
qA,B,C,D :- vA,B,C & vA,E,D & vB,F,D & vG,C,D
d
qA,B,C,D :- vA,B,C & vB,C,D & vB,A,D
Problem 7: 8 points A market-basket data set contains 10 items. For a particular
sample of the data, the set of all maximal frequent itemsets is precisely the set of all pairs
of items. How many itemsets that are subsets of these 10 items will be a part of the
negative border as used in Toivonen's Algorithm?
Explain your answer brie y.
6
Problem 8: 16 points Consider the following conjunctive queries with arithmetic:
: panic :- aX,Y & aY,X & X Y
Q1 : panic :- aA,B & aB,A & A6=B
Q2
We wish to check whether or not 1 2.
a Rewrite 1 and 2 as recti ed rules.
Q
Q
Q
Q
b What are all the containment mappings from the uninterpreted subgoals of
those of 1?
Q2
to
Q
c Write the statement about arithmetic that must be checked to verify that
d Is the condition of c true?
Explain brie y.
7
Q1
2.
Q
Problem 9: 10 points Suppose a Web graph is undirected, i.e. page points to page
i
if and only page points to page . Are the following statements true or false? Justify
your answers brie y.
a The hubbiness and authority vectors are identical, i.e for each page, its hubbiness is
equal to its authority.
j
j
i
b The matrix that we use to compute PageRank is symmetric; i.e.
for all and .
M
i
j
8
M i; j
=
M j; i