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Fall 2009
CS131 – Combinatorial Structures
Homework 7
Homework 7, due Nov 3
You must prove your answer to every question.
Do not rely only on the homework for exercise: there are several selfcheck ex
ercises of the easier kind in the book, try to solve them, too!
Problem 1
(CLRS 3.14)
.
(a) (5pts) Is 2
n
+
1
=
O
(2
n
)?
Solution.
2
n
+
1
=
O
(2
n
) since it is 2
·
2
n
.
(b) (5pts) Is 2
2
n
=
O
(2
n
)?
Solution.
The function 2
2
n
is not
O
(2
n
) since 2
2
n
/2
n
=
2
n
→∞
as
n
→∞
.
Problem 2.
For the following pairs of functions, show which one grows faster and
why.
(a) (5pts) 4
log
n
or
n
2
?
Solution.
They are equal, since 4
log
n
=
n
log4
=
n
2
.
(b) (5pts) (
f
(
n
/2))
3
or (
f
(
n
/3))
2
where
f
(
n
)
=
2
n
?
Solution.
(
f
(
n
/2))
3
=
2
n
·
(3/2)
±
2
n
·
(2/3)
=
(
f
(
n
/3))
2
.
(c) (5pt) 1
+
3
+
3
2
+···+
3
n
or 3
n
+
1
+
3
n
+
2
+···+
3
2
n
?
Solution.
We have 1
+
3
+
3
2
+···+
3
n
²
3
n
+
1
+
3
n
+
2
+···+
3
2
n
. Indeed, a geometric
series has the same rate of growth as its largest term, so 1
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This note was uploaded on 10/15/2011 for the course MATHS 100 taught by Professor Fredphelps during the Fall '11 term at Jordan University of Science & Tech.
 Fall '11
 FredPhelps
 Math

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