What is the difference between recursion and iteration?

Understanding recursion and iteration is important for any programmer. These concepts are at the heart of how we design algorithms and solve problems in computer science. Recursion helps us tackle complex problems by breaking them into smaller, more manageable pieces. On the other hand, iteration is about repeating tasks in an organized way.

In this blog, we’ll explore recursion and iteration, their working principles, their pros and cons, and some scenarios of when to use them. We’ll also examine some practical examples and common mistakes to avoid.

Recursion#

Recursion is a programming technique where a function calls itself to solve smaller instances of the same problem until it reaches a base case. This self-calling approach simplifies complex problems by breaking them into more manageable subproblems.

Recursion in the real world

Imagine standing between two parallel mirrors. When you look into one mirror, you see an infinite series of reflections of yourself. Each reflection is a smaller version of the previous one, continuing until it becomes too small to see. This is similar to how recursion works: each function call creates a smaller problem to solve, and these calls until they reach a trivial case that can be easily solved.

How does recursion work?#

In a recursive function, there are two main components:

• Base case: The condition under which the function stops calling itself. This prevents infinite recursion and provides a direct answer for the simplest instance of the problem.

• Recursive case: The part of the function that reduces the problem into smaller instances and calls itself with small data.

The following illustration shows the workflow of a recursive function:

Simple example: Factorial calculation

The factorial of a number $( n )$ $($denoted as $( n! ))$ is the product of all positive integers less than or equal to $( n )$. Thus, $n! = n \times (n-1) \times (n-2) \times (n-3) \dots 1$. For example, $5! = 5 \times 4 \times 3 \times 2 \times 1$.

Here’s how a recursive function works to calculate the factorial of a number:

Note: Factorials are only defined for positive integers. For negative numbers, the factorial is undefined.

Explanation:

• Base case: If $n$ is $0$ or $1$, return $1$.

• Recursive case: Multiply $n$ by the factorial of $n−1$.

Visual representation

The following illustration shows the steps of the recursive factorial function.

Types of recursion#

The following are the three different types of recursion:

• Direct recursion

• Indirect recursion

• Tail recursion

Direct recursion

In direct recursion, a function directly calls itself within its definition. This is the most straightforward form of recursion.

Example:

Indirect recursion

In indirect recursion, a function calls another function, which calls the original function. This creates a cycle of function calls.

Example:

Tail recursion

In tail recursion, the recursive call is the last operation in the function. This allows some compilers or interpreters to optimize the recursion, potentially converting it into an iterative loop to save memory.

Example:

Advantages of recursion#

The advantages of using recursion are as follows:

• Simplifies complex problems: Recursion can simplify the solution to complex problems by breaking them down into smaller, more manageable subproblems. This is particularly useful for problems that naturally fit into the divide-and-conquer approach.

• A natural fit for problems with recursive structure: Recursion is a natural fit for problems with an inherent recursive structure, such as tree traversals, graph traversals, and certain combinatorial problems.

Disadvantages of recursion#

The disadvantages of using recursion are as follows:

• Higher memory usage due to call stack: Recursion involves creating a new stack frame for each recursive call, which can lead to higher memory usage compared to iterative solutions. Each call adds a layer to the call stack, which can grow significantly for deep recursions.

• Risk of stack overflow: If the recursion depth is too high, it can exceed the maximum call stack size and cause a stack overflow error. This is a common issue with problems that require a large number of recursive calls.

• Performance considerations: Recursive solutions can sometimes be less efficient than their iterative counterparts due to the overhead of multiple function calls and the additional memory usage. This can result in slower execution times, especially for problems that can be solved more efficiently with iterative methods.

Importance of recursion in coding interviews#

Recursion is an important concept in coding interviews because it demonstrates a candidate’s ability to think recursively and solve complex problems by breaking them down into simpler subproblems. Many common interview questions, such as those involving tree and graph traversals, dynamic programming, and divide-and-conquer algorithms, rely on recursion. For those looking to deepen their understanding, the course “Recursion for Coding Interviews” offers comprehensive insights and practical examples to prepare effectively.

This course is available in three more languages to provide a personalized learning experience and enhance understanding.

Iteration#

In programming and mathematics, iteration is synonymous with loops, where a block of code is executed repeatedly until a specified condition is met or a certain number of iterations have been reached. It allows computers to solve complex problems by breaking them down into simpler, repeated steps. In broader terms, iteration is not limited to computing; it is a concept found in various aspects of life and problem-solving. It embodies the idea of gradual refinement or progress through repetition.

Iteration in the real world

In gardening, watering plants regularly is essential for their growth and health. In terms of iteration, watering plants regularly involves a repetitive process aimed at ensuring their growth and health over time.

How does iteration work?#

In an iterative process, such as in a loop or iterative function, there are fundamental components that dictate its behavior:

• Initialization: Begin with initializing variables or setting initial conditions required for the iterative process.

• Condition: Check a condition that determines whether the iteration should continue or stop.

• Body: Execute a set of instructions or operations that represent the core logic of the iteration.

• Update: Modify variables or states within the iteration that progress toward meeting the condition or completing the process.

The following illustration shows the workflow of an iterative function:

Loop constructs: For, While, Do-While loops

Iteration in programming involves repeating instructions until a specific condition is met. The primary constructs for iteration are:

• For loop: This is used when the number of iterations is known beforehand.

• While loop: This is used when the number of iterations is unknown, and the loop continues as long as a condition is true.

• Do-While loop: This is similar to the while loop but ensures that the loop is executed at least once before the condition is tested.

Simple example: Calculating factorial using a loop

The factorial of a number $( n )$ $($denoted as $( n! ))$ is the product of all positive integers less than or equal to $( n )$. Thus, $n! = n \times (n-1) \times (n-2) \times (n-3) \dots 1$. For example, $5! = 5 \times 4 \times 3 \times 2 \times 1$.

Here’s how loops work to calculate the factorial of a number:

Advantages of iteration#

The advantages of using an iterative solution are as follows:

• Generally more memory efficient: Iteration typically uses less memory than recursion because it doesn’t require additional stack frames for each iteration. Instead, it uses a single loop control variable and executes within the same function call frame.

• Easier to understand for simple problems: Loops are straightforward and easier to understand for simple, repetitive tasks. They follow a clear sequence of initialization, condition checking, body execution, and update, which makes the logic easy to follow.

• No risk of stack overflow: Unlike recursion, iteration doesn’t involve deep call stacks, eliminating the risk of stack overflow. This makes iteration a safer choice for problems that require many repetitions or involve large input sizes.

Disadvantages of iteration#

The disadvantages of using an iterative solution are as follows:

• Can be less intuitive for problems with a recursive nature: For problems that naturally fit into a recursive framework, such as tree traversals, graph traversals, or combinatorial problems, iterative solutions can be less intuitive. The recursive approach often mirrors the problem’s inherent structure, making it more logical and easier to implement.

• Sometimes results in more complex code: Iterative solutions can sometimes result in more complex and less readable code, especially for inherently recursive problems. Managing loop control variables, maintaining additional data structures (e.g., stacks or queues), and ensuring correct logic can lead to convoluted code compared to a recursive solution that expresses the problem more elegantly.

Comparing recursion and iteration#

The following table shows a basic comparison between recursion and iteration:

When to use recursion vs. iteration#

The following are some guidelines for choosing between recursion and iteration:

• Problem nature:

  • Use recursion for problems that naturally fit a recursive structure or require breaking down into smaller subproblems.

  • Use iteration for problems with a straightforward repetitive process or when the number of iterations is known beforehand.

• Complexity and readability:

  • Use recursion when it simplifies the code and makes it more readable.

  • Use iteration when recursion would make the code unnecessarily complex or when avoiding the risk of stack overflow is essential.

• Performance considerations:

  • Use recursion if the problem’s recursive nature allows for more elegant and maintainable code and manageable stack usage.

  • Use iteration if memory usage is a concern and the problem can be solved efficiently with loops.

Types of problems suited for recursion#

The following are the problems suited for recursion:

• Divide and conquer: Problems like merge sort, quicksort, and binary search where the problem is divided into smaller subproblems that are solved recursively.

• Tree and graph problems: Traversals (e.g., in-order, pre-order, post-order for trees) and pathfinding in graphs that naturally fit a recursive approach.

• Dynamic programming: Problems like the Fibonacci sequence, where recursion with memoization simplifies the solution.

• Combinatorial problems: Generating permutations and combinations and solving puzzles like the Tower of Hanoi.

Types of problems suited for iteration#

The following are the problems suited for Iteration:

• Simple loops: Problems like summing a list of numbers, iterating through arrays or lists, and simple counting problems.

• Linear problems: Iterative solutions for problems requiring a linear scan, such as finding an array’s maximum or minimum value.

• Repetitive tasks: Problems that require repeating a task a known number of times, such as printing numbers from 1 to N or iterating through data structures like arrays and linked lists.

Programming examples#

In the following programming examples, we will see how a specific problem can be solved using recursive and iterative approaches:

Fibonacci sequence#

1. Recursive approach

The Fibonacci sequence is defined as:

with the base cases:

Explanation: In the recursive approach, the function calls itself with the previous two terms until it reaches the base cases $(0 \space or \space 1)$. This method directly follows the mathematical definition of the Fibonacci sequence but can be inefficient for large $n$ due to repeated calculations.

2. Iterative approach

In the iterative approach, a loop is used to compute the Fibonacci sequence up to the $n^{th}$ term, maintaining two variables to store the last two Fibonacci numbers and updating them in each iteration.

Explanation: This method uses a for loop to iterate from $2$ to $n$, updating two variables that hold the previous two Fibonacci numbers. This approach is more memory-efficient and has a linear time complexity, making it suitable for large $n$.

Complexity analysis

Finding the maximum element in an array#

• Recursive approach
  The recursive approach involves dividing the array into smaller parts, finding the maximum element in each part, and combining the results.

• Iterative approach
  The iterative approach involves looping through the array of elements and keeping track of the maximum value found so far.

Complexity analysis

Final thoughts#

Both recursion and iteration are fundamental concepts in programming that offer different advantages and trade-offs. Both techniques will enhance your problem-solving abilities and deepen your understanding of algorithmic thinking.

What’s next?#

Recursion is a fundamental concept that often appears in coding interviews, particularly for positions at top tech companies. Practicing recursion is essential because it:

• Demonstrates problem-solving skills: Recursive solutions can showcase your ability to break down complex problems into simpler, more manageable subproblems.

• Highlights algorithmic thinking: Understanding and implementing recursive algorithms often requires a deep understanding of the underlying problem and its structure.

• Addresses common interview questions: Many classic interview problems, such as tree traversals, backtracking problems (e.g., generating permutations), and divide-and-conquer algorithms (e.g., quicksort, mergesort), rely heavily on recursion.

To further your knowledge and skills in recursion, we recommend the course Recursion for Coding Interviews. This course is offered in four languages: C++, Python, Java, and JavaScript.

Moving toward advanced levels: Dynamic programming#

Once you have a solid grasp of basic recursion, the next step is to delve into more advanced topics like dynamic programming (DP). DP is a powerful technique that optimizes recursive solutions by storing intermediate results, thereby avoiding redundant calculations and improving efficiency. It is widely used to solve complex problems in optimization, graph theory, and combinatorics.

Dynamic programming involves

• Identifying overlapping subproblems: Recognizing that a problem can be broken down into smaller, overlapping subproblems that can be solved independently.

• Using memoization or tabulation: To avoid redundant calculations, the results of subproblems are stored in a table (memoization) or filled up a table iteratively (tabulation).

• Optimizing recursive solutions: Transforming a naive recursive solution into a more efficient DP solution.

To further your knowledge and skills in dynamic programming, we recommend the course Grokking Dynamic Programming: A Deep Dive. This course is offered in four languages: C++, Python, Java, and JavaScript.