Comparing Functions

1. Comparing Functions

2. Notes on Notation N = {0,1,2,3, ... } N+ = {1,2,3,4, ... } R = Set of Reals R+ = Set of Positive Reals R* = R+ U {0}

3. Comparing f(n) and g(n) Let f be a function from N to R. O(f) (Big O of f) is the set of all functions g from N to R such that: 1. There exists a real number c>0 2. AND there exists an n0 in N Such that: g(n) £ cf(n) whenever n ³ n0

4. Notation and Pronunciation ProperNotation: g ÎO(f) “g is oh of f” Also Seen: g = O(f)

5. Big Omega Let f be a function from N to R. W(f) (Big W of f) is the set of all functions g from N to R such that: 1. There exists a real number c>0 2. AND there exists an n0 in N Such that: g(n) ³ cf(n) whenever n ³ n0

6. Big Theta Q(f) = O(f) ÇW(f) “g is of Order f” “g is Order f” g ÎQ(f)

7. Little o and Little Omega o(f) = O(f) - Q(f) w(f) = W(f) - Q(f)

8. English Interpretations • O(f) - Functions that grow no faster than f • W(f) - Functions that grow no slower than f • Q(f) - Functions that grow at the same rate as f • o(f) - Functions that grow slower than f • w(f) - Functions that grow faster than f

9. Limit Formulas if for some if or

10. More Limit Formulas if for some if

11. The Last Limit Formula if

12. Properties • Transitivity • if f Î O(g) and g Î O(h) then f Î O(h) • Same holds for Q , W, o, and w • Anti Symmetry (Sort of ...) • f Î O(g) if and only if g ÎW(f) • May replace O with o and W with w

13. Some More Properties • Symmetry • if f ÎQ(f), then g ÎQ(f) • Reflexivity • f Î O(f) • Also true for Q and W • Not True for o and w

14. And Even More Properties • Big Theta is an equivalence relation • f R g if and only if f ÎQ(g) • O(f+g) = O(max(f,g)) • Also true for Q and W