Remove K Digits: Optimal Solutions Using Stack and Queue Algorithms

Removing digits from a number to achieve the smallest possible value might sound simple, but the complexity escalates when optimizing for performance. The Remove K Digits problem is a classic algorithm challenge, often posed in coding interviews and competitive programming. Leveraging Stack and Queue algorithms can lead to highly efficient solutions. This blog dives into optimal strategies, ensuring you understand the theory and can apply it practically.

Understanding the Remove K Digits Problem

The Remove K Digits problem asks us to remove k digits from a given number to form the smallest possible number. For example:

• Input: num = "1432219", k = 3
• Output: "1219"

Here, removing the digits ‘4’, ‘3’, and ‘2’ results in the smallest number possible.

At first glance, one might think brute-force removal of all combinations would solve this, but that’s computationally expensive. This is where Stack and Queue algorithms shine, offering efficient and elegant solutions.

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Why Use Stack and Queue Algorithms?

Stacks and queues are fundamental data structures in computer science, known for their simplicity and efficiency:

• Stacks follow a Last-In-First-Out (LIFO) approach.
• Queues operate on a First-In-First-Out (FIFO) basis.

Their characteristics make them ideal for solving the Remove K Digits problem, allowing for dynamic decision-making and backtracking, which helps in constructing the smallest number possible.

Optimal Stack-Based Solution for Remove K Digits

Algorithm Overview

The stack-based approach is intuitive yet powerful. The main idea is to maintain a monotonic increasing stack to ensure the smallest digits stay while removing the larger ones.

Step-by-Step Stack Solution

1. Initialize an empty stack to store digits.
2. Iterate through each digit in the input number:
   • While the stack isn’t empty, the current digit is smaller than the stack’s top, and k > 0, pop the top of the stack (remove the larger digit).
   • Push the current digit onto the stack.
3. Handle remaining removals: If k is still greater than 0 after the iteration, remove the last k digits from the stack.
4. Build the final number: Concatenate the remaining digits and remove any leading zeros.

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Python Code for Stack-Based Solution

def removeKdigits(num: str, k: int) -> str:
    stack = []
    for digit in num:
        while stack and k > 0 and stack[-1] > digit:
            stack.pop()
            k -= 1
        stack.append(digit)

    # Remove remaining digits if k > 0
    while k > 0:
        stack.pop()
        k -= 1

    # Remove leading zeros
    result = ''.join(stack).lstrip('0')

    return result if result else "0"

Example Walkthrough

• Input: num = "1432219", k = 3
• Stack Process:
  • Add ‘1’ → Stack: ['1']
  • Add ‘4’ → Stack: ['1', '4']
  • ‘3’ < ‘4’, pop ‘4’, k=2 → Stack: ['1']
  • Add ‘3’ → Stack: ['1', '3']
  • ‘2’ < ‘3’, pop ‘3’, k=1 → Stack: ['1']
  • Add ‘2’ → Stack: ['1', '2']
  • ‘2’ == ‘2’, add ‘2’ → Stack: ['1', '2', '2']
  • ‘1’ < ‘2’, pop ‘2’, k=0 → Stack: ['1', '2']
  • Add ‘1’ → Stack: ['1', '2', '1']
  • Add ‘9’ → Stack: ['1', '2', '1', '9']

Final number: "1219"

Queue-Based Approach to Remove K Digits

Though stacks are more common for this problem, queues offer a unique perspective, particularly for problems requiring order preservation.

Queue Strategy

A queue can help in a greedy approach by always enqueueing the smallest digit found and dequeuing larger ones when necessary.

When to Use Queues?

• When processing input in a strict sequential order.
• When the problem requires handling a sliding window of digits.

While not as efficient as the stack method for Remove K Digits, queues can still offer clarity in specific variations of the problem.

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Time and Space Complexity Analysis

Stack-Based Solution:

• Time Complexity: O(n), where n is the number of digits in the input. Each digit is pushed and popped at most once.
• Space Complexity: O(n), for the stack storing digits.

Queue-Based Solution:

• Time Complexity: O(n), but might involve additional overhead in certain implementations.
• Space Complexity: O(n), similar to the stack solution.

Common Mistakes to Avoid

1. Not Handling Leading Zeros: Always remember to strip leading zeros in the final result.
2. Forgetting Edge Cases: If k equals the length of the number, the result should be "0".
3. Ignoring Remaining Removals: After the first iteration, ensure any remaining k digits are removed from the end.

FAQs

What is the Remove K Digits problem?
It’s an algorithm problem where you remove k digits from a number to make it the smallest possible.

Why use stacks for Remove K Digits?
Stacks efficiently handle dynamic removal of larger preceding digits, helping to maintain the smallest possible sequence.

Can queues be used for Remove K Digits?
Yes, though less common, queues can be applied, especially in variations of the problem requiring sequential processing.

What happens if k equals the number’s length?
The result should be "0" since all digits are removed.

How do you handle leading zeros?
Use string manipulation functions like lstrip('0') to remove leading zeros from the final result.

Is the stack approach faster than brute-force?
Yes, the stack approach is O(n) compared to the brute-force method’s exponential complexity.

Conclusion

The Remove K Digits problem is a brilliant exercise in optimizing algorithms using Stack and Queue data structures. The stack approach, in particular, provides an elegant and efficient solution, ideal for both interviews and real-world applications. Whether you’re preparing for a coding challenge or enhancing your algorithm skills, mastering this problem will undoubtedly sharpen your understanding of fundamental data structures.