Recursive Algorithm

Defining Recursive Algorithm

• \( a \geq 1 \) is the number of recursive calls
• \( b > 1 \) is the factor by which the problem size is divided
• \( f(n) \) represents the cost of the work done outside the recursive calls, which includes the cost of dividing the problem and the cost of merging the results

Recursive Algorithm Example

def factorial(n):
    if n == 1:
        return 1
    else:
        return n * factorial(n-1)

Advantages and Disadvantages of Recursive Algorithm

• Recursive algorithms make the code look clean and elegant.
• A complex task can be broken down into simpler sub-problems using recursion.
• Sequence generation is easier with recursion than using some nested iteration.

• Sometimes the logic behind recursion is hard to follow through.
• Recursive calls are expensive (inefficient) as they take up a lot of memory and time.
• Recursive functions are hard to debug.

Deciphering 3 Properties of Recursive Algorithm

In deeper explorations, there are three integral properties that characterise recursive algorithms. These include self-similarity, the base case, and the recursion rule.

Property 1: Self-Similarity

def fibonacci(n):
    if n <= 1:
       return n
    else:
       return (fibonacci(n-1) + fibonacci(n-2))

Property 2: Base Case

def factorial(n):
    if n == 1:
        return 1
    else:
        return n * factorial(n-1)

Property 3: Recursion Rule

def fibonacci(n):
    if n <= 1:
       return n
    else:
       return (fibonacci(n-1) + fibonacci(n-2))

Non Recursive Algorithm versus Recursive Algorithm

Define Non Recursive Algorithm

Comparing Non Recursive Algorithm and Recursive Algorithm

Practical Examples of Non Recursive Algorithm

def factorial(n):
    result = 1
    for i in range(1, n + 1):
        result *= i
    return result

def fibonacci(n):
    if n <= 1:
       return n
    a, b = 0, 1
    for _ in range(n):
        a, b = b, a + b
    return a

Deep Dive into Recursive Algorithm Definition

Breaking Down Recursive Algorithm Definition

Application of Recursive Algorithm in Computer Science

Recursive algorithms have profound implications and widespread applications in various realms of computer science, owing to their ability in presenting concise and clean solutions to intricate problems. With their unique problem-solving approach that deals with smaller instances of the same problem, recursive algorithms often prove to be immensely beneficial in tackling complex scenarios. Let's explore a few paramount applications of recursive algorithms:

1. Sorting Algorithms: Recursive algorithms drive some of the most efficient sorting algorithms in computer science, such as Merge Sort and Quick Sort. They utilise the divide-and-conquer strategy to divide the dataset into smaller subsets, recursively sort them, and finally reassemble them into a sorted whole.

2. Tree and Graph Data Structures: Recursive algorithms are extensively used in various operations involving tree and graph data structures. Be it performing Depth-First Search on a graph, or traversing a Binary Search Tree, recursive algorithms provide the simplest and the most intuitive solutions. The process of breaking the problem down to smaller sub-problems aligns with the inherent hierarchical structure of trees and graphs, making recursion the go-to approach for many tasks involving these data structures.

3. Dynamic Programming: Recursion plays a crucial role in dynamic programming, a method used for solving complex optimization problems by breaking them down into simpler sub-problems. Recursive algorithms aid in defining the optimal substructure of the problem, which forms the crux of dynamic programming.

4. Parsing and Tree-Based Computations: Recursive algorithms are of immense help in parsing expressions and executing tree-based computations. Recursive descent parsing, a common method used in writing compilers and interpreters, uses recursion to handle nested structures.

Remember that applications of recursive algorithms are not restricted to the ones listed. The potential extends to any problem that can be broken down into smaller, solvable units. Choosing between recursion and iteration depends heavily on the problem at hand and the computational resources available, making it pivotal to understand the strengths and weaknesses of both approaches.

Exploring Recursive Algorithm Examples

Simple Recursive Algorithm Example

def factorial(n):
    if n == 1:
        return 1
    else:
        return n * factorial(n - 1)

Complex Recursive Algorithm Example

class Node:
    def __init__(self, value):
        self.value = value
        self.left = None
        self.right = None

def dfs(node):
    if node is None:
        return
    print(node.value)
    dfs(node.left)
    dfs(node.right)

The function accepts a binary tree node as input, prints the value of the node, and then recursively calls itself for the left child, and then the right child of the node. It uses the nature of function call stacks to backtrace to previous nodes after reaching a leaf node, simulating the functionality of a depth-first search.

Real World Applications of Recursive Algorithms