CS 373: Combinatorial Algorithms, Spring 2001
Homework 0, due January 23, 2001 at the beginning of class
Name:
Net ID:
Alias:
Neatly print your name (first name first, with no comma), your network ID, and a short alias into
the boxes above.
Do not sign your name. Do not write your Social Security number.
Staple this sheet of paper to the top of your homework.
Grades will be listed on the course web site by alias give us, so your alias should not resemble your
name or your Net ID. If you don’t give yourself an alias, we’ll give you one that you won’t like.
This homework tests your familiarity with the prerequisite material from CS 173, CS 225, and
CS 273—many of these problems have appeared on homeworks or exams in those classes—primarily
to help you identify gaps in your knowledge.
You are responsible for filling those gaps on
your own.
Parberry and Chapters 1–6 of CLR should be sufficient review, but you may want to
consult other texts as well.
Before you do anything else, read the Homework Instructions and FAQ on the CS 373 course web
page (http://wwwcourses.cs.uiuc.edu/
∼
cs373/hw/faq.html), and then check the box below. This
web page gives instructions on how to write and submit homeworks—staple your solutions together
in order, write your name and netID on every page, don’t turn in source code, analyze everything,
use good English and good logic, and so forth.
I have read the CS 373 Homework Instructions and FAQ.
Required Problems
1.
(a) Prove that any positive integer can be written as the sum of distinct powers of 2. For
example: 42 = 2
5
+ 2
3
+ 2
1
, 25 = 2
4
+ 2
3
+ 2
0
, 17 = 2
4
+ 2
0
.
[Hint: ‘Write the number
in binary’ is
not
a proof; it just restates the problem.]
(b) Prove that any positive integer can be written as the sum of distinct
nonconsecutive
Fibonacci numbers—if
F
n
appears in the sum, then neither
F
n
+1
nor
F
n
−
1
will.
For
example: 42 =
F
9
+
F
6
, 25 =
F
8
+
F
4
+
F
2
, 17 =
F
7
+
F
4
+
F
2
.
(c) Prove that
any
integer (positive, negative, or zero) can be written in the form
∑
i
±
3
i
,
where the exponents
i
are distinct nonnegative integers. For example: 42 = 3
4
−
3
3
−
3
2
−
3
1
, 25 = 3
3
−
3
1
+ 3
0
, 17 = 3
3
−
3
2
−
3
0
.
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CS 373
Homework 0 (due 1/23/00)
Spring 2001
2. Sort the following 20 functions from asymptotically smallest to asymptotically largest, indi
cating ties if there are any. You do not need to turn in proofs (in fact, please
don’t
turn in
proofs), but you should do them anyway just for practice.
1
n
n
2
lg
n
lg
∗
n
2
2
lg lg
n
+1
lg
∗
2
n
2
lg
∗
n
⌊
lg(
n
!)
⌋
⌊
lg
n
⌋
!
n
lg
n
(lg
n
)
n
(lg
n
)
lg
n
n
1
/
lg
n
n
lg lg
n
log
1000
n
lg
1000
n
lg
(1000)
n
(
1 +
1
1000
)
n
n
1
/
1000
To simplify notation, write
f
(
n
)
≪
g
(
n
) to mean
f
(
n
) =
o
(
g
(
n
)) and
f
(
n
)
≡
g
(
n
) to mean
f
(
n
) = Θ(
g
(
n
)).
For example, the functions
n
2
,
n
,
(
n
2
)
,
n
3
could be sorted either as
n
≪
n
2
≡
(
n
2
)
≪
n
3
or as
n
≪
(
n
2
)
≡
n
2
≪
n
3
.
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 Spring '09
 A
 Dynamic Programming, Analysis of algorithms

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