Practice Questions
1. Which of the following are true?
(a) 3n = O(n
2
)
(b) 3n
2
= O(nlgn)
(c) nlgn =
Θ
(20000nlgn)
(d) 2
n
=
Ω
(n
1000
)
Answers: (a), (c), (d)
(a) n
2
clearly grows larger than 3n as n gets large.
(c) 20000 is a constant in front of nlgn
(d) All exponential functions grow faster than all polynomial
functions.
(b) is false because 3n
2
grows faster than nlgn.
2.
What is the runtime of the following segment of code in
terms of n? Give an upper bound and justify it. Only a tight
upper bound will be accepted as a correct answer.
int i=1;
while (i <= n) {
int j = i;
while (j > 0)
j = j/2;
i++;
}
The outer loop runs n times. The inner loop will run log i times
at most since there is repeated halving. Since i never exceeds n,
it is safe to say the inner loop runs at most log n times. Hence
an upper bound on the run time is O(nlgn).
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3. Determine the following sum:
∑

=
1
0
3
n
i
i
This is a geometric sum with a first term of 1, with a common
ratio of 3, with n terms. Here's the sum:
2
1
3
3
1
3
1
3
1
0

=


=
∑

=
n
n
n
i
i
.
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 Fall '09
 Computer Science, Exponential Functions, Harshad number, LG, upper bound

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