Question

Our goal for this question is to analyse the running time of the following function: 1 def nested(n): 2 """Assume n is an integer that is greater than 1.""" 3 b=1 4 while b <= n: # Loop 1 5 i=1 6 while i < b: # Loop 2 7 print(i) 8 i = i * 3 9 b = b + 1 (a) First, prove that for all functions $f, g: \mathbb{N} \to \mathbb{R}^+$ and $b \in \mathbb{R}^+$, if all three of the following conditions are true: (i) $g(n) \in O(f(n))$ (ii) $f(n)$ and $g(n)$ are eventually $\ge b$ [Correction: this used to say $\ge 1$] (iii) then $\log_b(g(n)) \in O(\log_b f(n))$. You may not use any external properties about Big-Oh; we're looking for you to prove this statement using the definition of Big-Oh.

          Our goal for this question is to analyse the running time of the
following function:
1
def nested(n):
2
"""Assume n is an integer that is greater than 1."""
3
b=1
4
while b <= n:
# Loop 1
5
i=1
6
while i < b:
# Loop 2
7
print(i)
8
i = i * 3
9
b = b + 1
(a) First, prove that for all functions $f, g: \mathbb{N} \to \mathbb{R}^+$ and $b \in \mathbb{R}^+$, if all three of the following conditions
are true:
(i) $g(n) \in O(f(n))$
(ii) $f(n)$ and $g(n)$ are eventually $\ge b$ [Correction: this used to say $\ge 1$]
(iii)
then $\log_b(g(n)) \in O(\log_b f(n))$.
You may not use any external properties about Big-Oh; we're looking for you to prove this statement
using the definition of Big-Oh.

Our goal for this question is to analyse the running time of the
following function:
1
def nested(n):
2
"""Assume n is an integer that is greater than 1."""
3
b=1
4
while b <= n:
# Loop 1
5
i=1
6
while i < b:
# Loop 2
7
print(i)
8
i = i * 3
9
b = b + 1
(a) First, prove that for all functions f, g: ℕ→ℝ^+ and b ∈ℝ^+, if all three of the following conditions
are true:
(i) g(n) ∈ O(f(n))
(ii) f(n) and g(n) are eventually ≥ b [Correction: this used to say ≥ 1]
(iii)
then (g(n)) ∈ O( f(n)).
You may not use any external properties about Big-Oh; we're looking for you to prove this statement
using the definition of Big-Oh.

Added by Richard A.

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