## What Is A Recursive Algorithm? With Examples And Exercises

# A Recursive Algorithm Is An Algorithm In Which A Function Calls Itself Recursively. In Other Words, The Function Is Called To Itself During Execution To Finally Arrive At The Final Response.

These algorithms are typically used to solve problems that divide more significant problems into smaller subproblems so they can be solved more easily.

A **recursive algorithm** consists of steps in which each step is calculated recursively. The algorithm divides a problem into minor issues at each stage to reach the next step. This process continues until the situation finally reaches its base state, which is the final answer or helps to achieve the definitive solution.

For example, you could write a recursive function to calculate the factorial of a number. Here, the factorial process to calculate the factorial of n calls itself with n-1 to arrive at the factorial of a smaller number and then calculates the factorial of n by multiplying it by n.

## Another example of a recursive algorithm

Another example of **a recursive algorithm** is calculating a number’s power. In this algorithm, we can easily calculate the power of a number by defining a return function. For example, suppose we want to calculate the nth power of integers a. In this case, we can define the following return function:

def power(a, n):

if n == 0:

return 1

elif n % 2 == 0:

return power(a, n//2) * power(a, n//2)

Otherwise:

return power(a, n//2) * power(a, n//2) * a

In this function, if the power of n is equal to zero, the value one is returned. Otherwise, if the power of n is odd, the process calls itself again with n-1 to make the power even and increase the value of a. If the power of n is even, the function calls itself again with n/2. It calculates the value of a to the power n/2 by multiplying them again. This algorithm recursively calculates the power of the number a using its halved and recursive capabilities.

## What is the difference between a recursive algorithm and an iterative algorithm?

**The recursive and iterative algorithms** are both computational algorithms that can be used to solve a problem, but they differ.

In the iterative algorithm, an iteration loop is used to solve a problem, which is solved iteratively in each step. In this case, the problem is solved iteratively in each stage, and the final result is obtained.

But in **the recursive algorithm**, the function is called to reach the final answer. In other words, the problem is recursively divided into smaller sub-problems at each stage, and by solving the sub-problems, we realize the solution to the whole situation.

In general, **the recursive algorithm** can be used instead of the iterative algorithm in many cases, but the iterative algorithm may be the best solution in some cases. Both algorithms may be different in terms of computational complexity, so to solve a problem, one should choose the best algorithm.

## Is the computational complexity of the recursive algorithm consistently higher than the iterative algorithm?

No, the computational complexity **of the recursive algorithm is not always higher than the iterative algorithm, and depending on the problem in question and how the algorithm is implemented, the computational complexity of the recursive algorithm** may be lower than the iterative algorithm in some cases.

Sometimes, using **a recursive algorithm** can be a simple and readable solution to solve problems. For example, **the recursive algorithm** for calculating the Fibonacci number is better than the iterative algorithm in some cases.

However, in some cases, **the recursive algorithm** can have more computational complexity than the iterative algorithm due to the more significant number of function calls and code repetition. In these cases, using an iterative algorithm may be the best option.

Therefore, to choose the best algorithm to solve a problem, one should pay attention to the nature of the problem and how to implement the algorithm and compare the computational complexity of both algorithms.

## Additional explanation about how to implement the recursive algorithm

Implementing **the recursive algorithm** means defining a function in the body of which the function itself is called recursively to reach the final answer. To implement a recursive algorithm, one must first check the correct definition of the recursive function and make sure that the function is somehow divided into smaller subproblems at each step to get the final answer.

Then, care must be taken that the return function is not called indefinitely. For example, in the problem of calculating the Fibonacci number using **the recursive algorithm**, calling the process for large values of n may be very time-consuming, and we may even encounter memory problems. Therefore, care must be taken to shrink the regression function sufficiently at each step to obtain the final answer.

Also, in some cases, using temporary memory to store response values can help improve the efficiency **of the recursive algorithm. **For example, in the problem of calculating the Fibonacci number, using a list to keep the answer values at each step can help reduce the number of function calls and improve the algorithm’s efficiency.

Finally, to implement **the recursive algorithm,** we can use a control structure combining conditionals and loops to get the final answer. For example, the if-else condition or the for and while loop is used to implement **the recursive algorithm.**

## Is it always better to use a recursive algorithm than other algorithms?

No, using **a recursive algorithm** is not always better than other algorithms, and in some cases, different algorithms may be the best solution. Each algorithm has advantages and disadvantages, and depending on the type of problem and its conditions, one will be a better choice than the other.

Using **the recursive algorithm** in some cases can be the best solution due to the simplicity and readability of the code as well as the ability to understand and modify it easily. For example, in many compound and string problems, **the recursive algorithm** can be the best solution.

But in some cases, other algorithms, such as the iterative algorithm, maybe the best solution due to higher efficiency and fewer function calls. For example, in many computational and algorithmic problems, the iterative algorithm is the best solution. Therefore, to choose the best algorithm to solve a problem, one should pay attention to the nature of the problem and how to implement the algorithm and compare the computational complexity of both algorithms.

## An example of a problem where the iterative algorithm has provided the best solution

One of the examples for which the iterative algorithm is usually the best solution is calculating the power of a number.

To calculate the power of a number, you can use the iterative algorithm that multiplies the number by itself at each step, reduces the power by one, and repeats this process until the power becomes zero. This algorithm is as follows:

def power(x, n):

result = 1

while n > 0:

if n % 2 == 1:

result *= x

x *= x

n //= 2

return result

This iterative algorithm has the least number of function calls and is significantly faster than the **recursive algorithm** for prominent exponents. **Of course, in some cases, the recursive algorithm** may also work well. Still, in general, the iterative algorithm is the best solution for calculating powers of numbers due to its faster speed and fewer function calls.

## Is the computational complexity of a recursive algorithm consistently higher than other algorithms?

No, the computational complexity **of the recursive algorithm** is not always higher than other algorithms. In some cases, the recursive algorithm can make programming more accessible and readable; in others, it can be more efficient than different algorithms.

For example, the recursive algorithm can be implemented naturally and readably in some computational problems, such as calculating the Fibonacci number. Of course, limitations, such as the memory limit and the number of function calls, may affect this issue.

On the other hand, in some computational problems, such as sorting, different algorithms, such as iterative algorithms, work more efficiently and with less computational complexity than the **recursive algorithm.**

Therefore, to choose the best algorithm to solve a problem, one should pay attention to the nature of the problem and how to implement the algorithm and compare the computational complexity of both algorithms.

## What is the use of a recursive algorithm?

**The recursive algorithm** can solve a series of problems. This algorithm is used in many issues as a problem-solving algorithm based on repetition. Some applications of the recursive algorithm are:

- Mathematical calculations: problems such as power calculation, factorial calculation, Fibonacci number calculation, etc., are solved using a
**recursive algorithm.** - Natural language processing: In natural language processing, we may need to find a pattern in a string.
**Recursive algorithms**can be helpful in this field. - Simulation and computer games: In simulation and computer games, recursive algorithms are often needed to simulate movements and calculate their positions.
- Graph review and computer graphics: Recursive algorithms are used in graph review and processing, as well as in the design of graphic algorithms.
- Artificial intelligence and machine learning: In artificial intelligence and machine learning,
**the recursive algorithm**can implement deep learning algorithms and artificial neural networks.

In general, **the recursive algorithm** is used in many problems, and due to the simplicity and readability of the code, and the ability to understand and modify it quickly, it is considered one of the most widely used algorithms in computer science.

## Types of return in the recursive algorithm

The recursive algorithm has two recursion types: linear and non-linear.

**1. Linear return:** In this type of return, the return function is called only once, and all calculations are performed as a linear chain of calls. For example, the factorial calculation algorithm is as follows:

def factorial(n):

if n == 0:

return 1

otherwise:

return n * factorial(n-1)

In this algorithm, the factorial function is called only once, and all calculations are performed as a linear chain of calls.

**2. Non-linear return:** In this type of return, the return function is called more than once and has different branches that form a return tree. For example, the algorithm for calculating the Fibonacci number is as follows:

def fib(n):

if n == 0:

return 0

elif n == 1:

return 1

otherwise:

return fib(n-1) + fib(n-2)

In this algorithm, the fib function is called more than once and has different branches that form a return tree.

According to the return type, various optimizations can be done in the return algorithm. For example, in the Fibonacci number calculation algorithm, by using the memoization technique, the non-linear return can be converted into a linear return, thereby improving the performance of the algorithm.

## What is the sequential recursion algorithm?

**A recursive sequence algorithm** is an algorithm used to solve recursive sequences. This algorithm is defined recursively, and at each step, the values of the sequence elements are recursively calculated from the importance of the previous aspects of the sequence.

For example, suppose we have the following recursive sequence:

a[0] = 1

a[1] = 1

a[n] = a[n-1] + 2*a[n-2]

In this sequence, each element equals the sum of twice the previous element and a second previous element of the sequence. To solve this sequence with the sequential **recursion algorithm, the following function can be used:**

def sequence(n):

if n == 0:

return 1

elif n == 1:

return 1

Otherwise:

return sequence(n-1) + 2*sequence(n-2)

#### In this function, using recursion, the value of each sequence element is recursively calculated from the values of the previous elements of the sequence. For example, to calculate the fifth element of the sequence, the sequence function needs two recursive steps to calculate its value:

sequence(5) = sequence(4) + 2*sequence(3)

= (sequence(3) + 2*sequence(2)) + 2*(sequence(2) + 2*sequence(1))

= (sequence(2) + 2*sequence(1)) + 2*(sequence(1) + 2*sequence(0)) + 2*(sequence(1) + 2*sequence(0))

= (1 + 2*1) + 2*(1 + 2*1) + 2*(1 + 2*1)

= 13

By using the sequence **recursion algorithm**, sequences with different patterns can be solved. However, in cases where the sequence length is considerable, the sequence recursion algorithm may be inefficient due to many function calls. Using other methods, such as dynamic programming, is better.