Stack applications (expression evaluation, backtracking) in Data Structures Theory - Time & Space Complexity

Time Complexity: Stack applications (expression evaluation, backtracking)

O(n)

Understanding Time Complexity

Stacks help solve problems like checking expressions or exploring choices step-by-step.

We want to know how the time needed grows as the input or problem size grows.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function evaluateExpression(expr) {
  let stack = [];
  for (let char of expr) {
    if (isOperand(char)) {
      stack.push(char);
    } else if (isOperator(char)) {
      let b = stack.pop();
      let a = stack.pop();
      let result = applyOperator(a, b, char);
      stack.push(result);
    }
  }
  return stack.pop();
}
    

This code evaluates a simple expression using a stack by processing each character once.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Looping through each character of the expression once.
• How many times: Exactly once per character, so as many times as the expression length.

How Execution Grows With Input

As the expression gets longer, the number of steps grows directly with its length.

Input Size (n)	Approx. Operations
10	About 10 steps
100	About 100 steps
1000	About 1000 steps

Pattern observation: Doubling the input roughly doubles the work needed.

Final Time Complexity

Time Complexity: O(n)

This means the time to evaluate grows in a straight line with the size of the expression.

Common Mistake

[X] Wrong: "Because there are nested operations, the time must be quadratic or worse."

[OK] Correct: Each character is processed once; the stack operations are quick and do not multiply the work.

Interview Connect

Understanding how stacks handle tasks like expression evaluation or backtracking shows you can manage step-by-step problem solving efficiently.

Self-Check

"What if the expression included nested parentheses requiring recursive evaluation? How would the time complexity change?"

Practice

1. What is the main purpose of a stack in data structures?

easy

A. To store data randomly

B. To store data in first-in, first-out order

C. To store data in last-in, first-out order

D. To store data permanently

Solution

1. Step 1: Understand stack behavior

   A stack stores data so the last item added is the first one removed, called last-in, first-out (LIFO).

2. Step 2: Compare options with stack definition

   To store data in last-in, first-out order matches the LIFO behavior, while others describe different data structures or incorrect behavior.

3. Final Answer:

   To store data in last-in, first-out order -> Option C

4. Quick Check:

   Stack = LIFO order [OK]

Hint: Remember stack = last item in, first item out [OK]

Common Mistakes:

• Confusing stack with queue (FIFO)
• Thinking stack stores data randomly
• Assuming stack stores data permanently

2. Which of the following is the correct way to push an element x onto a stack named s?

easy

A. s.push(x)

B. s.pop(x)

C. s.enqueue(x)

D. s.insert(x)

Solution

1. Step 1: Identify stack operations

   Common stack operations are push (add), pop (remove), peek (view top).

2. Step 2: Match operation to pushing element

   To add element x, use push. Pop removes element. Enqueue is for queues. Insert is not standard stack method.

3. Final Answer:

   s.push(x) -> Option A

4. Quick Check:

   Push means add element [OK]

Hint: Push adds, pop removes from stack top [OK]

Common Mistakes:

• Using pop to add elements
• Confusing queue methods with stack
• Using non-standard method names

3. Consider the expression (3 + 2) * (4 - 1). Using a stack to evaluate this expression step-by-step, what is the final result?

medium

A. 15

B. 9

C. 21

D. Error

Solution

1. Step 1: Evaluate inside parentheses first

   (3 + 2) = 5 and (4 - 1) = 3

2. Step 2: Multiply the results

   5 * 3 = 15

3. Final Answer:

   15 -> Option A

4. Quick Check:

   (3+2)*(4-1) = 15 [OK]

Hint: Calculate parentheses first, then multiply [OK]

Common Mistakes:

• Ignoring parentheses order
• Adding instead of multiplying
• Confusing subtraction sign

4. A program uses a stack to check if brackets in an expression are balanced. Which of the following errors would cause the program to incorrectly say brackets are balanced?

medium

A. Pushing opening brackets onto the stack

B. Not popping from stack when a closing bracket is found

C. Ignoring characters other than brackets

D. Checking stack is empty at the end

Solution

1. Step 1: Understand bracket checking with stack

   Opening brackets are pushed; closing brackets pop the matching opening bracket.

2. Step 2: Identify error impact

   If popping is skipped on closing bracket, stack keeps extra openings, so program misses imbalance.

3. Final Answer:

   Not popping from stack when a closing bracket is found -> Option B

4. Quick Check:

   Pop on closing bracket needed [OK]

Hint: Pop stack on closing bracket to check balance [OK]

Common Mistakes:

• Forgetting to pop on closing bracket
• Pushing closing brackets instead of opening
• Not checking stack emptiness at end

5. In backtracking algorithms, how does a stack help in exploring possible solutions?

hard

A. By preventing any repeated steps automatically

B. By storing all solutions at once for quick access

C. By sorting the solutions in ascending order

D. By remembering choices and allowing undoing steps when a path fails

Solution

1. Step 1: Understand backtracking process

   Backtracking tries choices step-by-step and undoes them if they lead to failure.

2. Step 2: Role of stack in backtracking

   Stack stores the choices made so far, allowing the algorithm to go back (undo) when needed.

3. Final Answer:

   By remembering choices and allowing undoing steps when a path fails -> Option D

4. Quick Check:

   Stack = remember and undo choices [OK]

Hint: Stack tracks choices to backtrack easily [OK]

Common Mistakes:

• Thinking stack stores all solutions simultaneously
• Assuming stack sorts solutions
• Believing stack prevents repeated steps automatically