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) ]

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) ]

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)

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

Template for O(n) proofs

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

Template for proving that O(n) is false

Example Prove that n2 is not O(n)

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

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