1. 1 Algorithms CSCI 235, Fall 2015 Lecture 4 Asymptotic Analysis II

2. 2 Last time... Say that f is:NotationApproximately Loosely little omega gf=  (g) f is way bigger than gf>g omega gf=  (g) f is at least as big as gf>=g theta gf=  (g) f is about the same as gf=g oh gf=O(g) f is at most as big as gf<=g little oh gf=o(g) f is way smaller than gf<g

3. 3 Relationships between O,o, , ,   (g)  (g)  (g)  (g)  (g) is a subset of bigger f smaller f

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

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

6. 6  is a subset of  and of O  (g) union  (g) is a subset of  (g)  (g) union  (g) is a subset of  (g)  (g)  (g)  (g)  (g)  (g)

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

8. 8 Symmetric relationships f =  (g) if and only if g = o(f) why? f =  (g) if and only if g = O(f) why? f =  (g) if and only if g =  (f) why?

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

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

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

12. 12 Helpful hints 1)Not every pair of functions is comparable 2)It may be easier to test for o(g) and  (g). Try these first and then try O,  and . 3)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  (g),  (g) or  (g). 4) When in doubt, graph the functions.