Need help figuring out whether this statement is correct

Question

I have been tasked with figuring out whether this statement is correct or not

Is the expression on the left equal to big theta of n^3?

From my understanding it would be true if we were talking big O, since n^3 is the fastest growing term, but I am not sure with big theta.

score 0 · Answer 1 · answered Mar 02 '19 at 13:53

0

A pointer towards the solution.

As explained in here big-theta means your function gets "squeezed" between big-O and big-Omega as n grows.

To prove that statement is correct you need to prove that the expression is equivalent to O(n^3) and Omega(n^3).

answered Mar 02 '19 at 13:53

score 0 · Accepted Answer · answered Mar 02 '19 at 14:19

Let

f(n) = 1/4 n^3 + n^2 log(n) + 17 n^2

then

f(n)/n^3 = 1/4 + log(n)/n + 17/n

so

lim f(n)/n^3 = 1/4 + 0 + 0 = 1/4

meaning that -ε < f(n)/n^3 - 1/4 < ε for n > n0. Take ε = 1/8. Then

1/4 - 1/8 < f(n)/n^3 < 1/8 + 1/4
1/8 < f(n)/n^3 < 3/8
n^3 < 8f(n) < 3n^3

Taking ϴ()

ϴ(n^3) <= ϴ(f(n)) <= ϴ(n^3)

because ϴ sees all multiplicative factors as 1.

2 Answers2