1. CSE 373, Copyright S. Tanimoto, 2002 Asymptotic Analysis -
Motivation
Big O notation
Big Ω notation
Big Θ notation
Examples
CSE 373, Copyright S. Tanimoto, Asymptotic Analysis -

2. CSE 373, Copyright S. Tanimoto, 2002 Asymptotic Analysis -
Motivation
Space and Time requirements for software depend on many factors
We need a way to characterize the performance of an algorithm independent of factors such as hardware platform, OS, compiler, and implementation details.
Big O, Big Ω, and Big Θ notation provide such a means.
CSE 373, Copyright S. Tanimoto, Asymptotic Analysis -

3. CSE 373, Copyright S. Tanimoto, 2002 Asymptotic Analysis -
Big O (Introduction)
If computer C1 running program A takes T1(n) = c1 n2 milliseconds to reverse a list of n numbers and computer C2 takes twice as long to run the same program then
T2(n) = c2 n2 = 2 c1 n2
It’s the same program in each case and so we would like to characterize the running times in a way that makes them equal.
O(T1(n)) = O(T2(n)) = O(n2)
CSE 373, Copyright S. Tanimoto, Asymptotic Analysis -

4. CSE 373, Copyright S. Tanimoto, 2002 Asymptotic Analysis -
Big O (Intuition)
To say that f(n) is O(g(n)) is to say that
f(n) is bounded from above by some positive constant times g(n) for all n other than some finite number of specific exceptional values of n.
CSE 373, Copyright S. Tanimoto, Asymptotic Analysis -

5. CSE 373, Copyright S. Tanimoto, 2002 Asymptotic Analysis -
Big O (Definition)
Suppose f and g are non-negative functions on the domain of natural numbers N = {0, 1, ... }
f : N  +
g : N  +
Then f(n) is said to be O(g(n)) if and only if
there exist constants c > 0 and n0 such that whenever n > n0, we have f(n)  c g(n)
CSE 373, Copyright S. Tanimoto, Asymptotic Analysis -

6. CSE 373, Copyright S. Tanimoto, 2002 Asymptotic Analysis -
Big O (Example)
Let f(n) = n(n-1)/2 and g(n) = n2
Then f(n) is O(g(n)) because we can take
c = 1 and n0 = 0, and then
(n > 0)
f(n) = n(n-1)/2 = ½ n2 - ½ n  1  n2 = g(n)
CSE 373, Copyright S. Tanimoto, Asymptotic Analysis -

7. CSE 373, Copyright S. Tanimoto, 2002 Asymptotic Analysis -
Big O (Example, cont.)
Note:
f(n) = ½ n2 - ½ n is also O(n3), O(n4), etc.
f(n) is not O(n1).
CSE 373, Copyright S. Tanimoto, Asymptotic Analysis -

8. CSE 373, Copyright S. Tanimoto, 2002 Asymptotic Analysis -
Big O (More Examples)
n is O(n) take c = 1, n0 = 0.
2n is O(n) take c = 2, n0 = 0
n is O(2n) take c = 1, n0 = 0
100 n is O(n) take c = 200, n0 = 2
e.g., when n=3:   200  3
CSE 373, Copyright S. Tanimoto, Asymptotic Analysis -

9. CSE 373, Copyright S. Tanimoto, 2002 Asymptotic Analysis -
Big Ω (Definition)
Suppose f and g are non-negative functions on the domain of natural numbers N = {0, 1, ... }
f : N   g : N  +
Then f(n) is said to be Ω(g(n)) if and only if
there exists a constant c > 0 such that for all but a finite number of values of n, we have f(n)  c g(n)
CSE 373, Copyright S. Tanimoto, Asymptotic Analysis -

10. CSE 373, Copyright S. Tanimoto, 2002 Asymptotic Analysis -
Big Ω (Intuition)
To say that f(n) is Ω(g(n)) is to say that
f(n) is bounded from below by some positive constant times g(n) for all n other than some finite number of specific exceptional values of n.
CSE 373, Copyright S. Tanimoto, Asymptotic Analysis -

11. CSE 373, Copyright S. Tanimoto, 2002 Asymptotic Analysis -
Big Ω (Examples)
n is Ω(n) take c = 1
2n is Ω(n) take c = 1
n is Ω(2n) take c = ½
0.001 n + 1 is Ω(n) take c = 0.001
e.g., when n=3:    3
CSE 373, Copyright S. Tanimoto, Asymptotic Analysis -

12. CSE 373, Copyright S. Tanimoto, 2002 Asymptotic Analysis -
Big Ω (Examples, cont.)
200 n is not Ω(n2)
As n gets larger, n2 will eventually overtake 200 n
CSE 373, Copyright S. Tanimoto, Asymptotic Analysis -

13. CSE 373, Copyright S. Tanimoto, 2002 Asymptotic Analysis -
Big Θ (Definition)
Suppose f and g are non-negative functions on the domain of natural numbers N = {0, 1, ... }
f : N   g : N  +
Then f(n) is said to be Θ(g(n)) if and only if
f(n) is O(g(n)) and f(n) is Ω(g(n))
CSE 373, Copyright S. Tanimoto, Asymptotic Analysis -

14. CSE 373, Copyright S. Tanimoto, 2002 Asymptotic Analysis -
Big Θ (Examples)
2n is O(n) and 2n is Ω(n) so 2n is Θ(n)
log2n is O(log n) and
log2n is Ω(log n) so
log2n is Θ(log n)
Since changing the base of a logarithm is equivalent to multiplying the log by a constant, we can ignore the base when writing O(log n),
Ω(log n), or Θ(log n)
CSE 373, Copyright S. Tanimoto, Asymptotic Analysis -