Introduction to Reverse Polish Notation

Reverse Polish Notation (RPN) is a mathematical notation in which every operator follows all of its operands. This format eliminates the need for parentheses that are required by infix notation. For example, the expression "3 + 4" in conventional notation would be written as "3 4 + " in RPN.

In software engineering interviews, understanding how to evaluate expressions in RPN can demonstrate your ability to work with stacks and understand postfix expression evaluation. This post will explore a LeetCode problem (LeetCode 150. Evaluate Reverse Polish Notation) where you are given an array of strings representing an RPN expression and are required to evaluate it.

You are given an array of strings tokens that represents an arithmetic expression in a Reverse Polish Notation.

Evaluate the expression. Return an integer that represents the value of the expression.

Note that:

• The valid operators are '+', '-', '*', and '/'.

• Each operand may be an integer or another expression.

• The division between two integers always truncates toward zero.

• There will not be any division by zero.

• The input represents a valid arithmetic expression in a reverse polish notation.

• The answer and all the intermediate calculations can be represented in a 32-bit integer.

Example 1:

Input: tokens = ["2","1","+","3","*"]
Output: 9
Explanation: ((2 + 1) * 3) = 9

Example 2:

Input: tokens = ["4","13","5","/","+"]
Output: 6
Explanation: (4 + (13 / 5)) = 6

Example 3:

Input: tokens = ["10","6","9","3","+","-11","*","/","*","17","+","5","+"]
Output: 22
Explanation: ((10 * (6 / ((9 + 3) * -11))) + 17) + 5
= ((10 * (6 / (12 * -11))) + 17) + 5
= ((10 * (6 / -132)) + 17) + 5
= ((10 * 0) + 17) + 5
= (0 + 17) + 5
= 17 + 5
= 22

Consideration of Different Approaches

There are a couple of ways to approach the evaluation of an RPN expression:

1. Using a Stack: This is the most straightforward approach. You iterate over each token and use a stack to store operands. When you encounter an operator, you pop the required operands from the stack, apply the operator, and push the result back onto the stack. This approach leverages the LIFO (Last In, First Out) nature of stacks and ensures that operations are carried out in the correct order.

2. Recursive Evaluation: Although not typical for this problem due to its complexity and overhead, a recursive solution would involve identifying operators and their corresponding operands and recursively solving smaller expressions. However, this is less efficient and can be more complex to implement than the stack approach.

For the purposes of software engineering interviews, the stack-based method is preferred due to its efficiency and the commonality of stack problems in coding interviews.

How to Solve the Problem

The key to solving the "Evaluate Reverse Polish Notation" problem efficiently lies in understanding the stack data structure. The overall time complexity of the stack-based solution is O(n), where n is the number of tokens, as each token is processed exactly once. Here’s a step-by-step breakdown of the approach:

1. Initialize an empty stack.

2. Iterate over each token in the array:

   • If the token is an operator (+, -, *, /), pop the top two elements from the stack, apply the operator, and push the result back onto the stack.

   • If the token is an operand, convert it to an integer and push it onto the stack.

3. After processing all tokens, the stack will contain one element, the result of the expression.

Python Solution

Here's the Python code with added comments for clarity:

class Solution:
    def resolve(self, a, b, operator):
        # Helper function to perform arithmetic operations
        if operator == '+':
            return a + b
        elif operator == '-':
            return a - b
        elif operator == '*':
            return a * b
        elif operator == '/':
            return int(a / b)  # Use int to truncate towards zero

    def evalRPN(self, tokens: List[str]) -> int:
        stack = []
        operands = {'+', '-', '*', '/'}
        for token in tokens:
            if token in operands:
                second = stack.pop()  # Always pop the second operand first
                first = stack.pop()   # Then the first operand
                result = self.resolve(first, second, token)
                stack.append(result)  # Push the result back onto the stack
            else:
                stack.append(int(token))  # Convert numeric string to integer and push onto stack
        return stack.pop()  # The last element on the stack is the result

Conclusion

Evaluating expressions in Reverse Polish Notation using a stack is a clear and efficient approach. By understanding and implementing this strategy, you not only master a common interview problem but also deepen your comprehension of stack operations, a crucial aspect of algorithm design.

This solution is robust, handles all edge cases mentioned in the problem description, and operates within optimal time complexity, making it a great example of interview-ready code.