CSCE 235
Handout
25: Homework Problems on Complexity
Assigned April 30, 2003
Due to time constraint, there will not be a homework
assignment on the last topic: Complexity.
This handout shows some examples of what the homework problems would
have been.
1. Show
that 
.
We
need to find c and 
such that
for all 
.
If
we try 
, then we have
.
.
Thus, if we set 
and 
, then we know that 
for all 
. Thus, we have shown
that 
.
2. Show
that 
.
We
need to find c and 
such that
for all 
.
We know that 
. So, now, we can try
to show that 
instead. (So, we don’t have to worry about that +13
term.)
.
, when 
(for the equation to
be defined)
Now,
here’s the trick. If 
, then 
, and thus 
. We can then pick 
. Thus if 
and 
, then 
for all 
. Thus, 
.
3. Show
that 
We
need to find 
, 
and 
such that
for all 
.
Dividing by 
, we get
.
holds for 
and 
; 
holds for 
and 
. Thus, if 
, 
, and 
, then 
for all 
. Thus, we have shown
that 
.
4. In each case, find the smallest integer k
such that 
is 
.
(a) 
(b) 
(c) 
(d) 
(a) 
. Given the
observation that if 
is 
and 
is 
, then
is 
, we know that 
is 
where 
. Note that 2 is the
highest power.
(b) 
. This is a little
bit tricky and requires some experience to solve. We know that 
for 
. Now, we also know
that 
. So, we know that 
!!! We have found the
highest power for the term. Thus, 
is 
where 
.
(c) 
. Now, this may look
complicated. But remember, divide and
conquer! We know that 
. That is, 
is a function of two
functions: 
and 
. The highest power
for the 
is 3, and thus 
is 
. Now we need to find
out about 
. We know that 
for 
. Thus, 
. So, 
is 
. Now, given the
observation that if 
is 
and 
is 
, then 
is 
, we know that 
is 
, where 
.
(d) 
. This is a little
bit tricky too. The trick is this. We know that 
! So, 
. Thus, 
is 
where 
.
5. Show
that n! is 
.
We know that 
. Or, 
. Since 
, we know that 
. So, 
. And we know that 
. Thus, we have shown
that n! is 
.
Based on
Prof. Chuck Cusack’s Notes and (Ross
and Wright 1988).