1. Big O notation f = O(g(n)) c g(n)
There are c>0, n0 so that for n>n0
f(n) ≤ c g(n)
c g(n): upper bound of f(n)
f(n) n0
[ Small o: for every c>0 there is n0 so that: f(n) ≤ c g(n) ]

2. Big Ω notation f = Ω(g(n)) f(n)
There are c>0, n0 so that for n>n0
c g(n) ≤ f(n)
c g(n): lower bound of f(n)
c g(n) n0
[ Small ω: for every c>0 there is n0 so that: c g(n) ≤ f(n) ]

3. Big Θ notation There are c1,c2>0, n0 so that for n>n0
c2 g(n)
There are c1,c2>0, n0 so that for n>n0
c1 g(n) ≤ f(n) ≤ c2 g(n)
c1 g(n): lower bound of f(n)
c2 g(n): upper bound of f(n)
f(n)
c1 g(n)

4. ~ notation
f(n)~g(n) f is equal to g asymptotically
f(N)
g(N)

5. Template for O(n) proofs

6. Examples 1. T(n) = (n+1)2 find upper bound for T(n)
2. Is n3 O(0.001n3) ?

7. Template for proving that O(n) is false

8. Example
Prove that n2 is not O(n)

9. Recursion Very often a problem is decomposed to smaller ones
Solution to subproblem is simpler
Solutions are combined iteratively

10. Examples T(N) = T(N-1) + N, N≥2, T(1)=1 T(N) = T(N/2)+1, N≥2, T(1)=1
Loops to eliminate one item
T(N) = T(N/2)+1, N≥2, T(1)=1
Halve the input
T(N) = T(N/2)+N, N≥2, T(1)=0
Halve the input but examine every item in the input
T(N) = 2T(N/2)+N, N≥2, T(1)=0
Halve the input and linear pass