Question

1. For each function defined below, find the value of k such that s(n) = O(nk). For part (a), justify your answer from the definitions of O, O, and by finding explicit constants that work, following the examples in Proofs 7.2.1 and 7.2.3 in the zyBook. Don't just refer to Theorem 7.2.2. For part (b), you do not need to find explicit constants, just explain why your answer is correct. (a) s(n) = (2n + 1)(5n2+1) (b) s(n) = nºt(n) + ns where t(n) = (nb) (Hint: answer in terms of b.)

Answer 1

Answer(a)

Big O Notaion:

's(n) = $\Theta$ (n^k)

s(n) = (2n +1) (5n² + 1)

s(n) = (2n +1) (5n² + 1)

s(n) = ( 2n x 5n² + 2n + 5n² + 1)

s(n) = ( 10 n³ + 2n + 5n² + 1) // major term is n³

s(n) = ( 10 n³ )

n^k $\leqslant$ ( 10 n³ )

n^k = O(n³ ) k =0,1,2,3   according to k values we can select constant C values satisfying equation

s(n) = O(n³ )

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Omega O Notaion:

's(n) = $\Theta$ (n^k)

s(n) = (2n +1) (5n² + 1)

s(n) = (2n +1) (5n² + 1)

s(n) = ( 2n x 5n² + 2n + 5n² + 1)

s(n) = ( 10 n³ + 2n + 5n² + 1) // major term is n³

s(n) = ( 10 n³ )

n^k $\geq$ ( 10 n³ )

n^k = O(n³ ) k =3,4,5,....... according to k values we can select constant C values satisfying equation

s(n) = $\Omega$ (n³ )

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Theta O Notaion:

's(n) = $\Theta$ (n^k)

s(n) = (2n +1) (5n² + 1)

s(n) = (2n +1) (5n² + 1)

s(n) = ( 2n x 5n² + 2n + 5n² + 1)

s(n) = ( 10 n³ + 2n + 5n² + 1) // major term is n³

s(n) = ( 10 n³ )

n^k = ( 10 n³ )

n^k = (n³ ) k =3   constant C₁ = 1 and C₂ =1

s(n) = $\Theta$ (n³ )

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Answer(b)

(b) s(n) = n³ t(n) + n⁵ where t(n) = (n^b)

s(n) = n³ t(n) + n⁵

s(n) = n³ x n^b+ n⁵  

• case 1: if b = 0,1,2 then

s(n) = n³ x n^b+ n⁵  

s(n) = O(n^₅ )  

• case 2: if b = 3,4,...... then

s(n) = n³ x n^b+ n⁵  

s(n) = O(n³ n^b )