CS300-03_Algorithm_Analysis - 3 Asymptotic Growth Rate How...

1
3. Asymptotic Growth Rate
How good is our measure of work done to compare algorithms ?
How precisely can we compare two algorithms using our measure of work ?
Measure of work
# of passes of a loop
# of basic operations
Time complexity
c
⋅
(measure of work)

2
For solving a problem P , suppose that two algorithms A
1
and A
2
need 10
6
n
and 5 n basic operations, respectively, i.e.
basic operations
A
1
10
6
n
A
2
5 n
Which one is better ?
What are their time complexities ?
A
1
10
6
n O( n )
A
2
5 n O( n )
Does this mean that A
1
( A
2
) is as good as A
2
( A
1
) ?
Well, ……
∴ Both # of basic operations (loops) and time complexity are
imprecise !!!

3
Now, suppose that algorithms A
1
and A
2
need the following amount of time:
time complexity
A
1
10
6
n O( n )
A
2
n
2
O( n
2
)
Which one is better ?
A
1
is better if n > 10
6
A
2
is better if n < 10
6
Then, why time complexity ?
Suppose that n → ∞
Then, n
2
grows much faster than 10
6
n . i.e.,
Under the assumption that n → ∞
A
1
is better than A
2
∴ Asymptotic growth rate !!!
∴ Time complexity (measure of work) compares and classifies algorithms by the
asymptotic growth rate.
∞ →
∞ →
n
n
n
6
2
10
lim
10
6
n
T ( n )

4
N = {0, 1, 2, …}
N
+
= {1, 2, 3, …}
R = the set of real numbers
R
+
= the set of positive real numbers
R
*
= R
+
∪ {0}
f : N → R
*
and g : N → R
*
g is:
Ω ( f ): g grows at least as fast as f .
θ ( f ): g grows as the same rate as f .
O( f ): g grows no faster than f .

5
Definition : Let f : N → R
*
. O( f ) is the set of functions,
g : N → R
*
such that for some c ∈ R
+
and some
n
0
∈ N , g ( n ) ≤ c ⋅ f ( n ) for all n ≥ n
0
.
O( f ) is usually called
“big oh of f ”, “oh of f ”, or “order of f ”.
Note: In other books,
g ( n ) = O( f ( n )) if and only if there exist two positive constants c and n
0
such
that | g ( n )| ≤ c ⋅ | f ( n )| for all n ≥ n
0
Under the assumption that f : N → R
*
and g: N → R
*
, two definitions have
a minor difference.
How to check
n
2
, 10
5
n
2
- n , n
2
+ 10
10
, 10
3
n
2
+ n - 1 ∈ O( n
2
)
Is 10
10
n ∈ O( n
2
) ?
) ( '
) ( '
lim
) (
) (
lim
rule s Hopital' L' By : note
) ( O ,
) (
) (
lim *
n f
n g
n f
n g
f g c c
n f
n g
n n
n
∞ → ∞ →
∞ →
=
∈ ⇒ ∈ = R
What is it ?

6
Definition : Let f : N → R
*
. Ω ( f ) is the set of functions,
g : N → R
*
such that for some c ∈ R
+
and some
n
0
∈ N , g ( n ) ≥ c
⋅
f ( n ) for all n ≥ n
0
.
Ω ( f ) is usually called
“big omega of f ” or “omega of f ”.
Note: In other books,
g ( n ) = Ω ( f ( n )) if and only if there exist two positive constants c and
n
0
such that | g ( n )| ≥ c
⋅
| f ( n )| for all n ≥ n
0
How to check
10
5
n , n
2
, n
2
+ 10 n + 10
6
∈ Ω ( n ).
Are they also Ω (log n ) ?