1. Big Omega, Theta
Defn: T(N) = (g(N)) if there are positive constants c and n0 such that T(N)  c g(N) for all N  n0 .
Lingo: “T(N) grows no slower than g(N).”
Defn: T(N) = (h(N)) if and only if T(N) = O(h(N)) and T(N) = (h(N)).
Big-Oh, Omega, and Theta establish a relative order among all functions of N.
Oct 8, 2001
CSE 373, Autumn 2001

2. Intuition, little-Oh
Defn: T(N) = o(p(N)) if T(N) = O(p(N)) and T(N)  (p(N)).
notation
intuition
O (Big-Oh) 
 (Big-Omega) 
 (Theta) =
o (little-Oh)
<
Oct 8, 2001
CSE 373, Autumn 2001

3. More about Asymptotics
Fact: If f(N) = O(g(N)), then g(N) = (f(N)).
Proof: Suppose f(N) = O(g(N)). Then there exist constants c and n0 such that f(N)  c g(N) for all N  n0.
Then g(N)  f(N) for all N  n0, and so g(N) = (f(N)).
Oct 8, 2001
CSE 373, Autumn 2001

4. More terminology Suppose T(N) = O(f(N)).
f(N) is an upper bound on T(N).
T(N) grows no faster than f(N).
Suppose T(N) = (g(N)).
g(N) is a lower bound on T(N).
T(N) grows at least as fast as g(N).
If T(N) = o(h(N)), then we say that T(N) grows strictly slower than h(N).
Oct 8, 2001
CSE 373, Autumn 2001

5. Style If f(N) = 5N, then f(N) = O(N5) f(N) = O(N3)
f(N) = O(N)  preferred
f(N) = O(N log N)
Ignore constant factors and low order terms.
T(N) = O(N), not T(N) = O(5N).
T(N) = O(N3), not T(N) = O(N3 + N2 + N log N).
Bad style: f(N)  O(g(N)). Wrong: f(N)  O(g(N)).
Oct 8, 2001
CSE 373, Autumn 2001

6. Facts about Big-Oh If T1(N) = O(f(N)) and T2(N) = O(g(N)), then
T1(N) + T2(N) = O(f(N) + g(N)).
T1(N) * T2(N) = O(f(N) * g(N)).
If T(N) is a polynomial of degree k, then T(N) = (Nk).
log k N = O(N), for any constant k.
Oct 8, 2001
CSE 373, Autumn 2001

7. Techniques Algebra ex. f(N) = N / log N, g(N) = log N.
same as asking which grows faster, N or log 2 N.
Evaluate .
limit is
Big-Oh relation
f(N) = o(g(N))
c  0
f(N) = (g(N)) 
g(N) = o(f(N))
no limit
no relation
Oct 8, 2001
CSE 373, Autumn 2001

8. Techniques, cont’d L’Hôpital’s rule: If and , then .
example: f(N) = N, g(N) = log N.
Use L’Hôpital’s rule.
f’(N) = 1, g’(N) = 1/N.
 g(N) = o(f(N)).
Oct 8, 2001
CSE 373, Autumn 2001