Computer Science 340
Reasoning about Computation
Review Session 1
Thursday, January 10, 2008
Problem 1
Scientists at the Princeton Genomics Institute have discovered the following process:
Given a test tube filled with DNA strands, it is possible to insert an enzyme that will connect
two end points of stands (possibly of the same strand), and then dissolve. Experiments show
every pair of endpoints is equally likely to be joined. If two endpoints of the same strand
connect, then a ”DNA cycle” is formed.
Suppose that
n
enzymes are inserted into a test tube with
n
strands. What is the expected
number of DNA cycles that will be formed?
Solution:
Assume that the
n
enzymes act one at a time.
Suppose there are
i
strands currently.
The action of a single enzyme reduces the number of strands by 1 and results in a cycle
with probability
1
2
i

1
. (Once you fix one end point that the enzyme acts on, there are 2
i

1
choices for the other end point. Of these, exactly one will result in the formation of a cycle.)
Let
X
i
be an indicator random variable for the event that a cycle is formed when
i
strands
remain in the test tube.
Then E[
X
i
] =
1
2
i

1
.
The expected number of cycles formed is
∑
n
i
=1
E[
X
i
] =
∑
n
i
=1
1
2
i

1
= Θ(log
n
).
Problem 2
A model recently proposed in social networks to model the effect of distance on social
relationships is the following: Consider a graph where the vertices are integer points (
x, y
)
such that
x
∈ {
0
, . . . , n

1
}
and
y
∈ {
0
, . . . , n

1
}
.
Place an edge between two points
(
x
1
, y
1
) and (
x
2
, y
2
) with probability
p
(
x
1
, y
1
, x
2
, y
2
) =
1
max(

x
1

x
2

,

y
1

y
2

)
α
,
where
α >
0 is a constant. Let
X
n
be the degree of the vertex (0
,
0). Give an asymptotic
expression for the expectation E[
X
n
], i.e.
find a function
f
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 Fall '07
 CharikarandChazelle
 Probability theory, Andrey Markov, Indicator function, Markov's inequality

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