Question

Find out how often the recursive version of the fib function calls itself. Keep a global variable fibcount and increment it once in every call to \(\mathrm{fib}\). What is the relationship between \(f i b(n)\) and fibcount?

Short answer

The fibcount is proportional to the Fibonacci sequence: fibcount for fib(n) is F(n+2) - 1.

Step-by-step solution

Understanding the Problem

We need to determine how often a recursive function (specifically the Fibonacci function) calls itself. This means figuring out how many times the function "fib" executes during the computation.

Setup the Recursive Function

Define a recursive function for Fibonacci. Typically, the function is defined as: \[ \text{fib}(n) = \text{fib}(n-1) + \text{fib}(n-2) \]for \( n > 1 \), with base cases \( \text{fib}(0) = 0 \) and \( \text{fib}(1) = 1 \).

Introduce the Global Counter

To track the number of function calls, declare a global variable, say "fibcount", initialized to zero. This variable will be incremented each time the Fibonacci function is called.

Modify the Function to Count Calls

Every time the Fibonacci function "fib" is called, increase "fibcount" by 1. The function body should start with the line: ``` global fibcount fibcount += 1 ```

Compute the Recursive Breakdown

Understand that for each call to \( \text{fib}(n) \) (where \( n > 1 \)), the function will also call \( \text{fib}(n-1) \) and \( \text{fib}(n-2) \). This recursive nature means the function call count grows rapidly as \( n \) increases.

Determine the Relationship

The number of function calls is related to the Fibonacci sequence itself. Specifically, for computing \( ext{fib}(n) \), the fibcount is \( F(n+2) - 1 \), where \( F \) is the Fibonacci sequence, because the call tree grows exponentially in a pattern similar to the Fibonacci numbers.

Key concepts

Fibonacci Sequence

The Fibonacci sequence is a sequence of numbers where each number is the sum of the two preceding ones. It usually starts with 0 and 1. For example, the sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, and so forth. Each term after the initial two is the sum of the previous two. This makes the sequence very predictable but also a challenge when calculated using a recursive algorithm because of its exponential growth rate.

The mathematical formula representing the Fibonacci sequence is:

• For any number greater than 1, the Fibonacci number at position 'n' is given by: \[F(n) = F(n-1) + F(n-2)\]
• With the base cases: \[F(0) = 0, F(1) = 1\]

Recursive calculations of the Fibonacci sequence involve breaking down the problem into smaller subproblems, reducing the input size at each step until the base cases are reached.

Global Variables

Global variables are variables that are declared outside any function in a program. This means they can be accessed and modified by any function within the same program.

In the context of counting recursive function calls, a global variable serves as a counter. In our example, the Fibonacci function uses a global variable 'fibcount' to record how many times it has been called.

• Global variables provide a way to share data between functions without passing them as parameters.
• However, they can introduce challenges related to debugging and maintaining the code, as they can be altered from anywhere within the program.
• To declare a global variable in Python, simply place the keyword 'global' before the variable name in your function, like so:
  ```python global fibcount ```
• You must initialize the global variable outside any function, usually at the very beginning of your program.
  ```python fibcount = 0 ```

Function Calls

Function calls are one of the fundamental operations in programming. When a function is called, the program executes the block of code within that function.

In recursive functions like fib for the Fibonacci sequence, the function calls itself to solve smaller portions of the problem:

• This process continues until base cases are met, such as \( F(0) = 0 \) and \( F(1) = 1 \).
• Each call uses the current arguments to compute the next set of values.
• Recursive function calls create a call stack that the computer keeps track of until each recursive branch is resolved.

Function calls can be expensive in terms of memory and processing time, especially if they lead to redundant calculations, as seen with naive recursive Fibonacci calculations. Effective use of function calls is crucial in building efficient algorithms, often requiring optimizations such as memoization to reduce unnecessary computation.