Introduction to Algorithms
21 February 2005
Massachusetts Institute of Technology
6.046J/18.410J
Professors Charles E. Leiserson and Ronald L. Rivest
Practice Quiz 1 Solutions
Practice Quiz 1 Solutions
Problem 1.
Recurrences
Solve the following recurrences by giving tight
Θ
notation bounds.
(a)
T
(
n
) = 3
T
(
n/
5) + lg
2
n
Solution:
By Case 1 of the Master Method, we have
T
(
n
) = Θ(
n
log
5
(3)
)
.
(b)
T
(
n
) = 2
T
(
n/
3) +
n
lg
n
Solution:
By Case 3 of the Master Method, we have
T
(
n
) = Θ(
n
lg
n
)
.
(c)
T
(
n
) =
T
(
n/
5) + lg
2
n
Solution:
By Case 2 of the Master Method, we have
T
(
n
) = Θ(lg
3
n
)
.
(d)
T
(
n
) = 8
T
(
n/
2) +
n
3
Solution:
By Case 2 of the Master Method, we have
T
(
n
) = Θ(
n
2
log
n
)
.
(e)
T
(
n
) = 7
T
(
n/
2) +
n
3
Solution:
By Case 3 of the Master Method, we have
T
(
n
) = Θ(
n
3
)
.
(f)
T
(
n
) =
T
(
n

2) + lg
n
Solution:
T
(
n
) = Θ(
n
log
n
)
. This is
∑
n/
2
i
=1
lg 2
i
≥
∑
n/
2
i
=1
lg
i
≥
(
n/
4)(lg
n/
4) =
Ω(
n
lg
n
)
. For the upper bound, note that
T
(
n
)
≤
S
(
n
)
, where
S
(
n
) =
S
(
n

1)+lg
n
,
which is clearly
O
(
n
lg
n
)
.
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6.046J/18.410J Practice Quiz 1 Solutions
Name
2
Problem 2.
True or False, and Justify
Circle
T
or
F
for each of the following statements, and briefly explain why.
The better your
argument, the higher your grade, but be brief. Your justification is worth more points than your
trueorfalse designation.
(a)
T F
If
f
(
n
)
does not belong to the set
o
(
g
(
n
))
, then
f
(
n
) = Ω(
g
(
n
))
.
Solution:
False. For example
f
(
n
) =
n
sin(
n
)
and
g
(
n
) =
n
cos(
n
)
.
(b)
T F
A set of
n
integers in the range
{
1
,
2
, . . . , n
}
can be sorted by R
ADIX
S
ORT
in
O
(
n
)
time by running C
OUNTING
S
ORT
on each bit of the binary representation.
Solution:
False. This results in
Θ(log
n
)
iterations of counting sort, and thus an
overall running time of
Θ(
n
log
n
)
.
(c)
T F
An adversary can construct an input of size
n
to force R
ANDOMIZED
M
EDIAN
to run in
Ω(
n
2
)
time.
Solution:
False. The
expected
running time of R
ANDOMIZED
M
EDIAN
is
Θ(
n
)
.
This applies to
any
input.
Problem 3.
Short Solution
Give
brief
, but complete, solutions to the following questions.
(a)
Consider any priority queue (supporting I
NSERT
and E
XTRACT
M
AX
operations) in
the comparison model. Explain why there must exist a sequence of
n
operations such
that at least one operation in the sequence requires
Ω(lg
n
)
time to execute.
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 Spring '08
 ErikDemaine
 Algorithms, Tim, Counting sort

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