Question

Consider the Fibonacci function: Fib (0) = 1 Fib (1) = 1 Fib (n) = Fib (n-2) + Fib (n -1) Use hand-tracing to compute the following values: a. Fib(3) b. Fib(4) c. Fib(5) Notice how much extra work is required to compute these values because we need to compute the same value, for example, Fib (2) many times.

Short answer

Fib(3) = 3; Fib(4) = 5; Fib(5) = 8.

Step-by-step solution

Start with Base Cases for Fib(3)

The base cases for the Fibonacci function are defined as \(Fib(0) = 1\) and \(Fib(1) = 1\). For computing \(Fib(3)\), we need the values of \(Fib(2)\) and \(Fib(1)\). Each of these lower values needs to be found first.

Hand-trace to Compute Fib(2)

Using the recursive formula \(Fib(n) = Fib(n-2) + Fib(n-1)\), for \(n=2\) we compute: \(Fib(2) = Fib(0) + Fib(1) = 1 + 1 = 2\).

Calculate Fib(3) Using Known Values

Now that we have \(Fib(2) = 2\) and \(Fib(1) = 1\), use the formula to compute \(Fib(3)\): \(Fib(3) = Fib(1) + Fib(2) = 1 + 2 = 3\).

Compute Fib(4) Starting with Required Values

For \(Fib(4)\), we need to compute \(Fib(3)\) and \(Fib(2)\) again. These values are already known from previous calculations: \(Fib(3) = 3\) and \(Fib(2) = 2\).

Calculate Fib(4) Using Known Values

Using the formula \(Fib(n) = Fib(n-2) + Fib(n-1)\), calculate \(Fib(4)\): \(Fib(4) = Fib(2) + Fib(3) = 2 + 3 = 5\).

Compute Fib(5) Starting with Required Values

For \(Fib(5)\), the values \(Fib(3)\) and \(Fib(4)\) are needed again. Using the earlier results: \(Fib(3) = 3\) and \(Fib(4) = 5\), we proceed.

Calculate Fib(5) Using Known Values

Finally, compute \(Fib(5)\) by using already computed values: \(Fib(5) = Fib(3) + Fib(4) = 3 + 5 = 8\).

Key concepts

Fibonacci Sequence

The Fibonacci Sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. In the context of this exercise, we defined our sequence a bit differently, starting with the base cases of 1 for both 0 and 1 indexes. It is defined as follows:

• Fib(0) = 1
• Fib(1) = 1
• Fib(n) = Fib(n-2) + Fib(n-1) for n > 1

This sequence appears in various fields of mathematics and science, playing significant roles in number theory and computer science. It helps us to understand the principles of recursion as each term is built from previous terms, showing the power of repetitive operations in algorithms.

Hand Tracing

Hand tracing involves manually going through the code or algorithm step by step to understand its execution better. This method is especially useful for recursive functions like the Fibonacci sequence. This way, we can visualize how recursive calls are made.
When computing values like Fib(3), Fib(4), and Fib(5), hand tracing guides us through each step:

• We start with our known base cases.
• Progress by computing higher numbers recursively.
• Keep track of intermediate results to avoid recomputations.

This helps students learn to predict the flow of recursive functions, understand intermediary calculations, and identify inefficiencies.

Base Cases

Base cases are critical in recursion as they define the simplest form of the problem and avoid infinite recursion. In the Fibonacci sequence, the base cases are Fib(0) = 1 and Fib(1) = 1.
These base cases provide the initial conditions that help in constructing the sequence. Without base cases:

• The recursion would not know when to stop.
• We could fall into an endless loop with the function calling itself indefinitely.
• Getting accurate computational results would be impossible.

Base cases terminate the recursive function by returning specific predefined values.

Algorithm Efficiency

Algorithm efficiency refers to the resources required for an algorithm to run effectively, in terms of time and space. For a recursive Fibonacci sequence, efficiency can become an issue. With each call, we recompute the same values numerous times, as seen in the exercise).
Inefficient algorithms might repeat computations, consuming more time than necessary. In the Fibonacci example, computing Fib(n) involves recalculating Fib(n-1) and Fib(n-2) multiple times, leading to redundant calculations.

• Reducing redundant computation can hugely boost efficiency.
• One way to improve it is using memoization - storing previously computed values.
• This approach reduces runtime from exponential to linear time.

By using such techniques, solving the Fibonacci sequence becomes both time-effective and a great lesson in optimizing recursive functions.