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Last Updated: Mar 31, 2024
Difficulty: Easy

Reverse Polish Notation

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Prerita Agarwal
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23 Jul, 2024 @ 01:30 PM

Introduction

Reverse Polish Notation, often abbreviated as RPN, is a mathematical notation where every operator follows all of its operands. It's a unique approach that does away with the need for parentheses to define operation order, making it a favorite in certain calculative processes and computer algorithms. 

Reverse Polish Notation

In this article, we'll learn what RPN is, how it works, and why it's useful in computing and mathematical operations. We'll explore its application through examples and look into how it simplifies complex expressions. 

Problem Statement

Let's talk about a common challenge in mathematics and computing: understanding and solving expressions. Usually, when we see math problems, they're written in a way most of us learned in school, like 2+3×4. But, there's a twist when it comes to Reverse Polish Notation (RPN). In RPN, the same expression is written as 234×+, which might look a bit odd at first glance.

The problem here is twofold. First, how do we read and make sense of expressions written in RPN? And second, how can we apply RPN to solve problems more efficiently, especially in computer algorithms and calculative processes where time and resources are precious?

To tackle these questions, we'll break down RPN into more digestible parts, look at examples, and go step-by-step through the process of evaluating RPN expressions. By understanding RPN, we can streamline computations and make algorithms simpler and faster, particularly in stack-based computing environments.

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Approach

Understanding Reverse Polish Notation (RPN) is all about recognizing patterns and following a set of simple rules. The core principle of RPN is that operators come after the operands they work on. This might sound a bit confusing at first, but it's actually straightforward once you get the hang of it. Let's break down the approach into clear steps to tackle RPN expressions efficiently:

  • Reading from Left to Right: Start at the beginning of the expression and move towards the end, one element at a time.
     
  • Handling Numbers: When you encounter a number, simply place it on a stack. Think of a stack as a pile of plates; you can only add or remove the top plate.
     
  • Dealing with Operators: Upon coming across an operator (like +, -, *, /), pop the two topmost numbers off the stack. These are the numbers the operator will work on.
     
  • Performing the Operation: Apply the operator to the two numbers you've just removed from the stack. For example, if you've popped 4 and 2 and the operator is *, you'll multiply them (2 * 4 = 8).
     
  • Storing the Result: Take the result of the operation and push it back onto the stack. This result may be used in subsequent operations as you continue through the expression.
     
  • Reaching the End: Once you've processed the entire expression, the final number left on the stack is the answer.
     

To make this approach clearer, let's walk through an example. Consider the RPN expression "3 4 + 2 * 7 /". We'll apply the steps outlined above to solve it:

  • Start with "3" and "4" and push them onto the stack.
     
  • Encounter "+", so pop "3" and "4", add them (3 + 4 = 7), and push "7" back.
     
  • Push "2" onto the stack, then encounter "*". Pop "7" and "2", multiply (7 * 2 = 14), and push "14".
     
  • Finally, push "7", encounter "/", and divide "14" by "7" to get "2", which is the answer.
     

This methodical approach simplifies handling expressions in RPN, making it a valuable tool for various mathematical and computational tasks.

Implementation

To bring Reverse Polish Notation (RPN) into practice, especially in programming, we need a clear strategy to evaluate expressions. We'll use Python for our examples, as it's known for its readability & is widely taught in colleges. Here's a step-by-step guide to creating a simple RPN calculator:

Step 1: Set Up the Stack

First, we need a stack to hold our numbers. In Python, we can use a list for this purpose.

stack = []

Step 2: Split the Expression

Assuming our RPN expression is a string, we'll split it into individual elements.

expression = "3 4 + 2 * 7 /"  # Example RPN expression
tokens = expression.split()  # Split by spaces

Step 3: Process Each Token

We'll iterate through each token in the expression. If it's a number, we push it to the stack. If it's an operator, we pop the necessary numbers off the stack, perform the operation, and push the result back onto the stack.

for token in tokens:
    if token in ['+', '-', '*', '/']:
        # Pop two numbers from the stack
        num2 = stack.pop()
        num1 = stack.pop()


        # Perform the operation and push the result back
        if token == '+':
            stack.append(num1 + num2)
        elif token == '-':
            stack.append(num1 - num2)
        elif token == '*':
            stack.append(num1 * num2)
        elif token == '/':
            stack.append(num1 / num2)  # Assuming num2 is not 0
    else:
        # If it's a number, push it onto the stack
        stack.append(float(token))  # Convert to float for division

Step 4: Retrieve the Result

After processing all tokens, the result of the RPN expression is the last item left on the stack.

result = stack.pop()
print(f"The result is: {result}")

 

Now look into another example to learn the implementation -: 

  • Python

Python

def evaluate_rpn(expression):
# This function evaluates an RPN expression
# 'expression' is a list of strings, each being either an operand or an operator

# Initialize an empty stack
stack = []

# Define the operations
operations = {
'+': lambda a, b: a + b,
'-': lambda a, b: a - b,
'*': lambda a, b: a * b,
'/': lambda a, b: a / b
}

# Iterate over each element in the expression
for token in expression:
if token in operations:
# If the token is an operator, pop two operands from the stack
b = stack.pop()
a = stack.pop()
# Perform the operation and push the result back onto the stack
operation = operations[token]
stack.append(operation(a, b))
else:
# If the token is a number, push it onto the stack
stack.append(float(token))

# The final element in the stack is the result
return stack.pop()

# Example usage
expression = "3 4 + 2 * 7 /".split()
result = evaluate_rpn(expression)
print(f"The result is: {result}")

Output

The result is: 2.0


In this implementation, we start by defining a function evaluate_rpn that takes a list of strings as its input. Each string is either a number (an operand) or a mathematical operator. We use a stack (represented by a list in Python) to keep track of numbers.

As we iterate over the expression, we check if the current token is an operator. If it is, we pop the two most recent numbers off the stack, perform the operation, and push the result back onto the stack. If the token is a number, we simply convert it to a float and push it onto the stack. After processing all tokens, the last number on the stack is the result of the expression.

Approach – Space Optimized

When working with Reverse Polish Notation (RPN) in programming, space efficiency becomes a key consideration, especially for devices with limited memory. Optimizing the use of space can significantly enhance the performance of your applications. Here's how we can tweak our previous RPN calculator implementation to be more space-efficient:

Reusing the Stack

Instead of creating new variables for each operation, we can directly manipulate the values within the stack. This reduces the number of temporary variables we need.

In-place Operations

Whenever possible, perform operations in-place. For example, instead of popping two numbers off the stack, operating on them, and then pushing the result back, we can pop just one number, perform the operation with the top of the stack, and update the top of the stack with the result.

Limiting Precision

Floating-point numbers can take up more space due to their precision. In cases where high precision is not critical, consider limiting the number of decimal places.

Here's a revised version of the RPN calculator that incorporates these space optimizations:

  • Python

Python

def evaluate_rpn_optimized(expression):
stack = []

operations = {
'+': lambda a, b: a + b,
'-': lambda a, b: a - b,
'*': lambda a, b: a * b,
'/': lambda a, b: float(a) / b, # Ensure division results in a float
}

for token in expression:
if token in operations:
b = stack.pop()
# Perform the operation with the new top of the stack and the popped value
stack[-1] = operations[token](stack[-1], b) # In-place operation
else:
# Convert token to float and limit precision if needed
stack.append(float(f"{float(token):.2f}")) # Example of limiting to 2 decimal places

return stack[0] # The result remains at the top of the stack

# Example usage
expression = "3 4 + 2 * 7 /".split()
result = evaluate_rpn_optimized(expression)
print(f"The optimized result is: {result:.2f}") # Limiting the output precision

Output

The optimized result is: 2.00

 

This optimized approach ensures that the RPN calculator uses memory more efficiently, making it suitable for environments where resources are scarce. The changes are subtle but contribute to a more streamlined and efficient codebase.

Frequently Asked Questions

What is Reverse Polish Notation (RPN)?

RPN is a mathematical notation where every operator follows all of its operands. It eliminates the need for parentheses to determine the order of operations, making it straightforward and efficient, especially in computing.

Why is RPN used in computing?

RPN aligns well with the way computers process data, particularly with stack-based operations. It simplifies parsing and reduces the computational overhead associated with traditional infix notation, leading to quicker and more efficient calculations.

How can I practice RPN?

A great way to practice is by converting regular expressions into RPN and vice versa. You can also implement an RPN calculator, like the examples provided in this article, to get hands-on experience with the concept.

Conclusion

In this article, we've explored Reverse Polish Notation (RPN), a fascinating and efficient way to represent mathematical expressions. We started by understanding the basic premise and problem statement surrounding RPN, then moved on to a detailed approach for interpreting and evaluating RPN expressions. Through clear examples and straightforward Python implementations, we've demonstrated how RPN works in practice and how it can be optimized for space efficiency.

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Topics covered
1.
Introduction
2.
Problem Statement
3.
Approach
4.
Implementation
4.1.
Step 1: Set Up the Stack
4.2.
Step 2: Split the Expression
4.3.
Step 3: Process Each Token
4.4.
Step 4: Retrieve the Result
4.5.
Python
5.
Approach – Space Optimized
5.1.
Reusing the Stack
5.2.
In-place Operations
5.3.
Limiting Precision
5.4.
Python
6.
Frequently Asked Questions
6.1.
What is Reverse Polish Notation (RPN)?
6.2.
Why is RPN used in computing?
6.3.
How can I practice RPN?
7.
Conclusion