Stack Applications: Expression Conversion And Evaluation

Stacks are very useful in expressions like:

A + B * C

They help with:

• expression conversion
• expression evaluation
• matching brackets
• recursion and function calls

Types Of Expressions

Infix

Operator is between operands.

A + B

Prefix

Operator comes before operands.

+ A B

Postfix

Operator comes after operands.

A B +

Why Prefix And Postfix Are Helpful

They avoid confusion about brackets and operator priority.

For example:

A + B * C

In postfix:

A B C * +

Now the order is clear.

Infix To Postfix

Main Idea

1. Read expression left to right
2. Send operands directly to output
3. Put operators on stack
4. Pop higher-priority operators before pushing lower-priority ones
5. Pop all remaining operators at the end

Algorithm

infixToPostfix(expr):
    create empty stack
    for each symbol in expr:
        if symbol is operand:
            add to output
        else if symbol is '(':
            push to stack
        else if symbol is ')':
            while top of stack is not '(':
                pop from stack to output
            pop '('
        else if symbol is operator:
            while stack not empty and precedence(top) >= precedence(symbol):
                pop from stack to output
            push symbol

    while stack not empty:
        pop from stack to output

Complexity

Time: O(n)

Why:

• each symbol is pushed or popped at most once

Space: O(n)

Infix To Prefix

One common method:

1. reverse infix expression
2. swap ( and )
3. convert to postfix
4. reverse result

Complexity

Time: O(n)

Space: O(n)

Postfix Evaluation

Main Idea

1. Read from left to right
2. If operand, push it
3. If operator, pop two operands
4. apply operation
5. push result back

Algorithm

evaluatePostfix(expr):
    create empty stack
    for each symbol in expr:
        if symbol is operand:
            push symbol
        else:
            b = pop()
            a = pop()
            result = a operator b
            push result
    return pop()

Example

2 3 4 * +

Process:

push 2
push 3
push 4
* => 3 * 4 = 12
push 12
+ => 2 + 12 = 14

Answer: 14

Complexity

Time: O(n)

Space: O(n)

Prefix Evaluation

Main Idea

Read from right to left.

If operand, push it.

If operator, pop two values, apply operator, push result.

Algorithm

evaluatePrefix(expr):
    create empty stack
    for each symbol from right to left:
        if symbol is operand:
            push symbol
        else:
            a = pop()
            b = pop()
            result = symbol applied to a and b
            push result
    return pop()

Complexity

Time: O(n)

Space: O(n)

Other Applications Of Stacks

• function call management
• undo and redo
• browser backtracking
• depth-first search
• balancing parentheses

Quick Summary

Task	Time	Space
Infix to Postfix	O(n)	O(n)
Infix to Prefix	O(n)	O(n)
Postfix Evaluation	O(n)	O(n)
Prefix Evaluation	O(n)	O(n)