Question

a) Prove that running time T(n)=n3+30n+1 is O(n3) [1 mark]

b) Prove that running time T(n)=(n+30)(n+5) is O(n2) [1 mark]

c) Count the number of primitive operation of algorithm unique1 on page 174 of textbook, give a big-Oh of this algorithm and prove it. [2 mark]

d) Order the following function by asymptotic growth rate [2 mark]

a. 4nlogn+2n

b. 210

c. 3n+100logn

d. n2+10n

e. n3 f. nlogn

Answer 1

f(n) = O(g(n)) means there are positive constants c and n0, such that 0 ≤ f(n) ≤ cg(n) for all n ≥ n0

n^3+30n+1 is O(n^3)

=>  n^3+30n+1 <= c(n^3)

Let's assume c = 2
=>  n^3+30n+1 <= c(n^3)
=>  n^3+30n+1 <= 2(n^3)
=>  30n+1 <= n^3

n^3 is greater than 30n+1 for all n >= 6

so, n^3+30n+1 is O(n^3) for all c = 2 and n0 = 6