Maximum Number Possible by Doing at-Most K Swaps

authorImageVarun Saharawat29 Apr, 2026
Maximum Number Possible by Doing at-Most K Swaps

The largest number in k swaps problem asks you to make the biggest number you can by swapping digits no more than k times. A greedy approach, where you always swap with the biggest digit, doesn't always work, especially when there are repeated digits. We use recursion and backtracking to try different swaps and see which one gives us the best result.

The Problem in Largest Number in K Swaps

The goal is straightforward: you are given a positive integer (as a string) and an integer K. You need to find the maximum number you can create by swapping any two digits at most K times. The "at most" part is vital. If you reach the maximum possible value in fewer than K swaps, you don't need to continue. However, the order of swaps matters immensely. A greedy choice, simply swapping the current digit with the largest one found later in the string, can sometimes fail if that large digit could have been better used for a different position.

Why Greedy Fails in Largest Number in K Swaps

A simple greedy idea is the following:  Always swap the current digit with the largest digit on the right. This works in some cases but fails when:
  • The same largest digit appears multiple times
  • Choosing one position blocks a better swap later
So, greedy is not reliable for this problem.

Backtracking Approach for Largest Number in K Swaps

Since a greedy method might miss the optimal solution, we use backtracking. This allows the algorithm to try every possible swap and "undo" it if it doesn't lead to the best result.

How the Algorithm Works:

  1. Base Case: If K becomes 0 or we reach the end of the string, stop.
  2. Find the Maximum: For the current position, identify the largest digit present in the remaining part of the string.
  3. Perform Swaps: If the largest digit found is greater than the current digit, swap them.
  4. Recurse: Move to the next position with K-1 swaps remaining.
  5. Backtrack: Swap the digits back to their original positions to explore other potential paths.

Steps to Solve Largest Number in K Swaps

  1. Base Case
    • If k = 0, stop
    • If you reach the end, stop
  2. Find Maximum Digit
    • Look at the remaining digits
    • Find the largest one
  3. Swap
    • If the largest digit is bigger than the current digit, swap
  4. Recurse
    • Move to the next position
    • Reduce k by 1
  5. Backtrack
    • Undo the swap
    • Try other possibilities
  6. Track Maximum
    • Keep updating the best number found

Largest Number in K Swaps Python

Python makes this problem easier to implement.
  • Convert the string into a list
  • Use a recursive function
  • Keep track of the maximum number
  • Only swap when the next digit is larger
Python is simple for recursion and helps you focus on logic. Also Read:
  1. Randomized Algorithm
  2. Geometric Algorithms
  3. Introduction To Red-Black Tree
  4. Advance Data Structure And Algorithms
  5. Heap Data Structure

Largest Number in K Swaps Java

In Java, we handle swaps using arrays.
  • Use char[] or StringBuilder
  • Store the maximum result in a variable
  • Use compareTo() for comparison
  • Always backtrack after recursion
The logic is the same, but the syntax is different.

Largest Number in K Swaps JavaScript

JavaScript uses arrays for this problem.
  • Convert string to array
  • Use a helper recursive function
  • Update result when a larger number is found
  • Avoid unnecessary swaps
Again, the logic stays the same. Read more:
  1. Introduction To Priority Queue
  2. Matrix Data Structure: Definition, Types & Applications
  3. Deque Data Structure: Definition, Operations & Applications
  4. Recursive Algorithms: Definition, Examples & Key Concepts
  5. Divide And Conquer Algorithm: Definition, Examples & Time Complexity

Largest Number in K Swaps Approaches

Let;s look at these approaches at a glance:
Feature Greedy Approach Backtracking (Optimal)
Accuracy Often misses the global maximum Always finds the correct maximum
Complexity O(N) O(N^K) but optimized with pruning
Use Case Simple cases with unique digits General cases with duplicate digits
Efficiency Very fast Slower but precise

Limitations and Edge Cases in Largest Number in K Swaps

You should handle these cases carefully:
  • Duplicate digits: Try all positions of the max digit
  • k = 0: No change in number
  • Large k: You may reach the answer early
  • Already maximum: No swaps needed
  • Leading zeros: Can appear after swaps
  • Single digit: No swaps possible

Time Complexity of Largest Number of K Swaps

  • Worst case: O(N^K)
  • Faster in practice due to skipping useless swaps

Space Complexity of Largest Number of K Swaps

  • Depends on recursion depth (k)
  • Extra space for storing maximum result
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FAQs

1. Why does the greedy approach fail in the largest number of k swaps problem?

The current step is the only focus of the greedy strategy. It selects the largest digit to the right in an attempt to perform the best quick switch. However, if the same greatest digit appears more than once, this could cause issues. Selecting the incorrect event could obstruct a subsequent, better arrangement. By investigating all feasible swap choices rather than making an early commitment, backtracking resolves this issue.

2. What is the time complexity of the largest number of k swaps solutions?

In the worst case, the time complexity is O(N^K), where N is the number of digits and K is the number of swaps that can be made. This is because there are many ways to swap at each step. But in real life, the solution runs faster because of pruning. We only do swaps when they make the number better, which cuts down on unnecessary recursive calls.

3. Why is backtracking the best approach for the largest number of k swaps?

Backtracking is a good method because it looks at all the possible options and makes sure that the final answer is the highest possible number. It doesn't depend on local choices like greedy methods do. It also lets us undo swaps and try different paths, which helps us find the global maximum.

4. Can dynamic programming be used for the largest number of k swaps problem?

Dynamic programming isn't the best way to solve this problem. The main reason is that the state, or the way the digits are arranged right now, is always changing and hard to store well. Storing and reusing results is more difficult and less efficient than backtracking because each arrangement can be very different.

5. How do you handle very large values of k when finding the largest number with k swaps?

If k is very large (for example, greater than or equal to the number of digits minus one), you can just put the digits in order from highest to lowest. You can do this because you have enough swaps to put the number in the best possible order without having to go all the way back.