Question Name

Question Name#

Problem#

A clear and concise statement of the problem.

Example#

Small example(s) to illustrate the problem.

Intuition#

Brief explanation of the initial thought process for solving the problem.

Assumptions#

List of any assumptions made in the problem solving process.

Constraints#

What are the Constraints for?#

Explanation of the constraints and their impact on the problem and solution.

Test Cases#

Set of test cases for validating the solution.

Edge Cases#

Discussion of any potential edge cases in the problem.

Walkthrough / Whiteboarding#

Detailed walkthrough of the problem-solving process.

Theoretical Best Time Complexity#

Discussion of the theoretical best time complexity for this problem.

Theoretical Best Space Complexity#

Discussion of the theoretical best space complexity for this problem.

Space-Time Tradeoff#

Analysis of the tradeoff between space and time complexity for the problem.

Solution (Potentially Multiple)#

Intuition#

Brief explanation of the core ideas or insights that form the basis for the solution.

Visualization#

Visual representation of the problem and solution (if applicable).

Algorithm#

fib(n):
    if n = 0 then     // base case
        return 0
    elseif n = 1 then // base case
        return 1
    else
        return fib(n - 1) + fib(n - 2)
    endif

Claim#

Claim: The algorithm, fib(n) is correct and returns the \(n\)th Fibonacci number.

(Proof by Strong Induction)

Proof#

https://cs.stackexchange.com/questions/14025/prove-correctness-of-recursive-fibonacci-algorithm-using-proof-by-induction

The intuition is that we assume that the algorithm is correct for all values of \(k \leq n\) and then prove that it is correct for \(n+1\). The distinction between strong and weak induction is that in strong induction, we assume that the algorithm is correct for all values of \(k \leq n\) whereas in weak induction, we assume that the algorithm is correct for \(k=n\).

Why do we need to use strong induction here? Because we need to look back at two previous values of \(k\) to compute the \(k+1\)th value. So, weak induction would not work here.

In certain problems like this one, the property at a certain step \(n\) depends not just on the property at the step \(n-1\), but potentially on multiple or even all previous steps. This is where strong induction becomes useful.

In the case of the Fibonacci sequence, \(F_{k+1} = F_k + F_{k-1}\), to prove the property for \(k+1\), we need the property to hold for both \(k\) and \(k-1\) steps, not just \(k\).

So in such situations where a step depends on multiple prior steps, we use strong induction because it allows us to assume that the statement holds for all previous steps, which provides a stronger base for the inductive step. This ‘looking backwards more’, as you phrased it, is indeed why we often use strong induction.

Base Case: for inputs \(0\) and \(1\), the algorithm returns \(0\) and \(1\) respectively. So this is correct.

Induction Hypothesis: Here, we assume that the algorithm defined as fib(k) is correct and returns the correct Fibonacci number for all values of \(k \leq n\), where \(n,k\in \mathbb{N}\).

We must now show that the algorithm is correct for \(n = k+1\). That means show that

\[ F_{k+1} = F_k + F_{k-1} \]

where \(F_k\) is the \(k\)th Fibonacci number.

Inductive Step:

  1. let fib(k) be true for all values until \(n\), this means that fib(k) correctly computes \(F_k\) for \(k = 0,1,2,\ldots,n\).

  2. From the induction hypothesis, we know fib(k) correctly computes \(F_k\) and fib(k-1)correctly computes \(F_{k-1}\)

  3. So, we can then say the below:

    \[\begin{split} \begin{align*} \text{fib(k+1)} &= \text{fib(k) + fib(k-1)} \\ &\text{(by definition of the Fibonacci function)} \\ \\ &= F_k + F_{k-1} & \\ \\ &= F_{k+1} \\ &\text{(By definition of Fibonacci numbers)}\\ \end{align*} \end{split}\]

  4. Thus by rules of mathematical Inducion, fib(n) always returns the correct result for all values of \(n\).

Implementation#

Code implementation of the algorithm.

Tests#

Set of tests for validating the algorithm.

Time Complexity#

Analysis of the time complexity of the solution.

Space Complexity#

Input Space Complexity#

Analysis of the space complexity of the input.

Auxiliary Space Complexity#

Analysis of the space complexity excluding the input and output space.

Total Space Complexity#

Analysis of the total space complexity of the solution.

References and Further Readings#

Any useful references or resources for further reading.