The image presents an exercise where you are asked to show that the recurrence relation $T(n) = T\left(\left\lfloor \frac{n}{2} \right\rfloor\right) + 1$ is $O(\log n)$, meaning it grows logarithmically. The attempts in the image try to solve this using two different approaches:

Key Points:

1. Attempt 1:

   • The goal is to prove that $T(n) \leq c \log n$ for some constant $c$.
   • The steps try to bound the recurrence relation by substituting $T(n)$ with a logarithmic expression.
   • The step where $\log(n+1)$ appears doesn't successfully reduce the problem, and hence, the attempt fails (as marked with the red face).

2. Attempt 2:

   • In this attempt, the approach is to subtract a constant $\alpha$, assuming $\alpha \in \mathbb{R}$, so that $T(n) \leq c \log(n - \alpha)$.
   • The key idea is to use a stronger assumption: $T(n) \leq c \log(n - \alpha)$, for some constant $\alpha$, where $\alpha \geq 1$.
   • After simplification, the expression bounds the recurrence as desired, leading to a logarithmic bound.
   • A base case check is suggested to finalize the proof.

Summary of Approaches:

• First Attempt: Unsuccessful due to an incorrect reduction step when trying to simplify the expression involving $\log(n+1)$.
• Second Attempt: Successful, provided a constant $\alpha$ is subtracted, and the base case holds.

Detailed Steps to Show $T(n) = O(\log n)$:

The recurrence $T(n) = T\left(\left\lfloor \frac{n}{2} \right\rfloor\right) + 1$ suggests that for each step, we reduce the size of $n$ by half. This is characteristic of logarithmic time behavior.

1. Base Case: $T(1) = 1$, which is constant and fits within $O(\log n)$.
2. Recurrence Analysis: $T(n) \leq T\left(\frac{n}{2}\right) + 1.$ Unfolding the recurrence: $T(n) = T\left(\frac{n}{2}\right) + 1 = T\left(\frac{n}{4}\right) + 1 + 1 = \dots = T(1) + \log_2 n = \log_2 n.$ Therefore, $T(n)$ is bounded by $O(\log n)$.

Would you like further details on any specific step or concept here?

Related Questions:

1. What is the general method for solving recurrence relations using the master theorem?
2. How can we prove that a recurrence is $O(\log n)$ using substitution methods?
3. What are some common pitfalls when trying to bound recurrences?
4. How can we verify the correctness of a base case in recursive algorithms?
5. Can the recurrence $T(n) = T\left(\frac{n}{3}\right) + 1$ also be $O(\log n)$?

Tip:

When solving recurrence relations, it's important to carefully check the base case to ensure that the derived asymptotic behavior holds across all $n$.