1. Algorithms CSCI 235, Spring 2019 Lecture 4 Asymptotic Analysis II

2. Last time... Say that f is: Notation Approximately Loosely
little omega g f=w(g) f is way bigger than g f>g
omega g f=W(g) f is at least as big as g f>=g
theta g f=Q(g) f is about the same as g f=g
oh g f=O(g) f is at most as big as g f<=g
little oh g f=o(g) f is way smaller than g f<g

3. Relationships between O,o,Q,W,w
is a subset of
w(g)
bigger f
W(g)
Q(g)
smaller f
O(g)
o(g)

4. W and w if f = w(g) then f = W(g) Why? Definition of w(n):
If f=W(g) is it necessarily true that f=w(g)?

5. O and o if f = o(g) then f = O(g) Why? Definition of o(n):
If f=O(g) is it necessarily true that f=o(g)?

6. Q is a subset of W and of O w(g) union Q(g) is a subset of W(g)
o(g) union Q(g) is a subset of O(g)
w(g)
O(g)
W(g)
Q(g)
o(g)

7. Q is the intersection of O and W
In other words:
f=Q(g) if and only if f=O(g) and f= W(g)
why?

8. Symmetric relationships
f = w(g) if and only if g = o(f)
why?
f = W(g) if and only if g = O(f)
f = Q(g) if and only if g = Q(f)

9. Example 1 What is the relationship between f and g?
What happens if you change the coefficients?

10. Example 2
What is the relationship between f and g?

11. Example 3 Can we have a function that is in O(g) but not o(g) or Q(g)?
Consider:
n if n is odd
1 if n is even
g(n) = 1
b) g(n) = n

12. Helpful hints Not every pair of functions is comparable
It may be easier to test for o(g) and w(g). Try these first and then try O, W and Q.
Sometimes you can deduce several relationships from the knowledge of only 1. For example: if a function is o(g) it is also O(g), but never Q(g), W(g) or w(g).
4) When in doubt, graph the functions.