Q.Write an algorithm to convert an infix expression into equivalent prefix expression using stack.

This is a plain theory question — it asks for an algorithm, not a code implementation. The answer must give the step-by-step procedure that a human (or a program) would follow, with the reasoning behind each step.

Algorithm: Infix to Prefix Conversion Using Stack

The core idea is to reverse the infix expression, swap parentheses, convert the result to postfix (which is easier with a stack), and then reverse again. This works because postfix and prefix are mirror images when the expression is reversed and parentheses are flipped.

Step 1: Reverse the infix expression.

For each character in the reversed string, swap ( with ) and vice versa. This reversed-and-swapped expression is what we will feed into a standard postfix conversion algorithm.

Step 2: Define operator precedence (highest to lowest):

• ^ (exponentiation) — right-associative
• *, / — left-associative
• +, - — left-associative

Step 3: Scan the reversed-and-swapped expression from left to right.

For each token:

• If it is an operand (letter, digit, or variable), append it directly to the output string.
• If it is a left parenthesis (, push it onto the stack.
• If it is a right parenthesis ), pop operators from the stack and append them to output until a ( is encountered. Discard the (.
• If it is an operator, then:
  • While the stack is not empty and the top of the stack has greater or equal precedence than the current operator (and for ^, equal precedence is not popped because it is right-associative), pop and append.
  • Push the current operator onto the stack.

Step 4: After the scan, pop any remaining operators from the stack and append them to the output.

The output at this point is the postfix form of the reversed expression.

Step 5: Reverse the entire output string.

The result is the required prefix expression. …

Why This Algorithm Works

The core idea is elegant: prefix notation (Polish notation) places operators before their operands. Since we read expressions left-to-right, converting directly to prefix is awkward because the operator comes before operands we haven't seen yet. The trick is to reverse the problem.

We reverse the infix expression, swap parentheses (opening becomes closing and vice versa), then run a modified infix-to-postfix algorithm. Finally, we reverse the postfix result to get prefix. This works because postfix places operators after operands — reversing the input and output effectively flips the operator placement.

The Algorithm Steps

Step 1: Reverse the infix expression

• Reverse the character order of the input string.
• Swap every ( with ) and every ) with (.

Step 2: Convert the reversed expression to postfix (using a stack)

• Scan the reversed expression left to right.
• If character is an operand (letter/digit), add it to output.
• If character is (, push onto stack.
• If character is ), pop from stack to output until ( is encountered (discard the ().
• If character is an operator:
  • While stack is not empty AND top of stack has greater or equal precedence than current operator, pop to output.
  • Push current operator onto stack.
• After scanning, pop all remaining operators from stack to output.

Step 3: Reverse the postfix expression to get prefix

Algorithm in Pseudocode

Algorithm infixToPrefix(infix):
    Input: infix expression as string
    Output: prefix expression as string

    1. reversed ← reverse(infix)
    2. for i from 0 to length(reversed)-1:
           if reversed[i] == '(':
               reversed[i] ← ')'
           else if reversed[i] == ')':
               reversed[i] ← '('

    3. postfix ← infixToPostfix(reversed)   // using modified precedence
    4. prefix ← reverse(postfix)
    5. return prefix

Python Implementation

def precedence(op):
    """Return precedence value of operator."""
    if op == '^':
        return 3
    elif op in ('*', '/'):
        return 2
    elif op in ('+', '-'):
        return 1
    return 0

def is_operator(ch):
    """Check if character is an operator."""
    return ch in ('+', '-', '*', '/', '^')

def infix_to_postfix(expression):
    """Convert infix to postfix using stack."""
    stack = []
    output = []
    
    for ch in expression:
        # Operand: add to output
        if ch.isalnum():  # alphanumeric (letters or digits)
            output.append(ch)
        # Opening bracket: push to stack
        elif ch == '(':
            stack.append(ch)
        # Closing bracket: pop until opening bracket
        elif ch == ')':
            while stack and stack[-1] != '(':
                output.append(stack.pop())
            stack.pop()  # remove '('
        # Operator: handle precedence
        elif is_operator(ch):
            while (stack and stack[-1] != '(' and 
                   precedence(stack[-1]) >= precedence(ch)):
                # For right-associative '^', use > instead of >=
                if ch == '^' and precedence(stack[-1]) == precedence(ch):
                    break
                output.append(stack.pop())
            stack.append(ch)
    
    # Pop remaining operators
    while stack:
        output.append(stack.pop())
    
    return ''.join(output)

def infix_to_prefix(infix):
    """Convert infix expression to prefix."""
    # Step 1: Reverse the infix expression
    reversed_infix = infix[::-1] …