CS237. Practice Problems Set 5: Randomized Min Cut Algorithm.
Random Variables
Ungraded. Please complete by Thurs Feb 24.
March 8, 2011
Reading.
M&U Chapter 1.4, L&L Notes on Random Variables. (See also Schaum’s Chapter
5.2,5.3,5.10,5.11, but the other books are more important this week.)
Optional Extra Practice.
M&U Ex 1.231.25
Exercise 1.
Consider the mincut algorithm given in class and M&U 1.4.
1. Prove by giving an explicit counterexample that a cut set
C
for graph
G
, does not have to
be a cut set for the graph obtained in the
i
th iteration of the algorithm.
2. The opposite however is true: every cutset of a graph in an intermediate iteration is a cutset
of the original graph.
Solution:
Consider the graph
G
= (
V, E
)
, where
V
=
{
1
,
2
,
3
,
4
,
5
}
and
E
=
{
1

3
,
1

2
,
2

4
,
2

3
,
3

4
,
3

5
,
4

5
}
. Consider the graph obtained by contracting
3
and
4
and contracting
1
and
3
. Consider further a cutset
C
=
{
1

3
,
1

4
,
1

2
}
. Then,
C
is a cut for
G
but not the for the intermediate graph since it does not contain the edge
1

3
at all. For the second part, consider a graph with some nodes contracted and let
C
be its cutset. Let us remove the
edges from the graph that are in
C
. We get at least two components of the intermediate graph. Rollback the contraction
operations. We will get at least two components of the original graph
G
. Therefore,
C
is a cutset of
G
as well.
Exercise 2.
Consider the following variation of the randomized min cut algorithm discussed in
class. Start with a graph of
n
vertices; contract it down to
m
vertices. Make
k
copies of this
m
vertex graph and now run the randomized algorithm on each copy of the smaller graph, each with
independent randomness. Finally, output the smallest cut set (found from one copy of the smaller
graph). Bound the probability of finding the minimum cutset.
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 Spring '11
 Goldberg
 Probability distribution, probability density function, Randomness, Cumulative distribution function, ﬁrst roll

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