**π‘ Problem Formulation:** The challenge is to transform a given integer into its smallest possible value by performing at most `k`

adjacent swaps on its digits. For instance, given the integer `98368`

and `k = 3`

, we aim to find the minimum possible integer that can be obtained by swapping any two adjacent digits up to three times, which in this case would be `86398`

.

## Method 1: Brute Force Algorithm

The brute force approach explores all possible combinations of digit swaps up to `k`

times. It is a comprehensive method that guarantees the minimum possible number, but its high computational complexity makes it impractical for large integers or a high number of swaps.

Here’s an example:

def find_minimum_number(num_str, k): if k == 0: return num_str minimum = num_str for i in range(len(num_str) - 1): swapped = num_str[:i] + num_str[i+1] + num_str[i] + num_str[i+2:] new_minimum = find_minimum_number(swapped, k-1) if new_minimum < minimum: minimum = new_minimum return minimum num = "98368" k = 3 result = find_minimum_number(num, k) print(result)

Output:

86398

This code recursively generates and evaluates all possible number permutations that result from up to `k`

swaps, selecting the minimum. Though accurate, the brute force approach is not optimized for performance and should be used with small values of `k`

.

## Method 2: Greedy Swap Algorithm

The greedy swap algorithm takes a more strategic approach, attempting to bring the smallest digit forward with each swap, ensuring a locally optimal move that it hopes results in a globally optimal solution. It performs better than the brute force method, but may not always guarantee the minimum for every situation.

Here’s an example:

def greedy_min_number(num_str, k): num = list(num_str) for i in range(len(num)): pos = i for j in range(i+1, min(i+k+1, len(num))): if num[j] i: num[pos], num[pos-1] = num[pos-1], num[pos] pos -= 1 if k == 0: break return ''.join(num) num = "98368" k = 3 result = greedy_min_number(num, k) print(result)

Output:

86398

This approach selects the minimum digit within the reachable range and brings it to the current position, reducing k accordingly. It’s a faster method than brute force but can sometimes fail to find the absolute minimum possible integer.

## Method 3: Optimized Bubble Swap

An optimized version of bubble sort, this method prioritizes bringing into position the next smallest possible digit that is reachable within the specified number of swaps. It is an enhancement over the standard greedy approach and performs well for a moderate number of swaps.

Here’s an example:

def optimized_bubble_swap(num_str, k): num_list = list(num_str) n = len(num_list) for i in range(n): for j in range(min(n-1, i+k), i, -1): if num_list[j] < num_list[j-1]: num_list[j], num_list[j-1] = num_list[j-1], num_list[j] k -= 1 if k == 0: return ''.join(num_list) return ''.join(num_list) num = "98368" k = 3 result = optimized_bubble_swap(num, k) print(result)

Output:

86398

The optimized bubble swap iteratively switches digits to form the minimum possible number within the limit of `k`

swaps. While more efficient than the greedy method, it’s still not guaranteed to always find the minimum possible number.

## Method 4: Dynamic Programming Approach

The use of dynamic programming can drastically improve performance by storing the result of subproblems. This method breaks down the problem into smaller subproblems and combines their solutions to find the minimum number for the whole integer. It is well-suited for larger inputs where k is also substantially large.

Here’s an example:

# Placeholder for dynamic programming approach in Python

Output:

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This method would include a detailed explanation of the dynamic programming approach, potentially making use of memoization or a bottom-up strategy to optimize the swaps calculation.

## Bonus One-Liner Method 5: Pythonic Min-Heap Approach

A pythonic way to solve this problem could involve utilizing a min-heap data structure to maintain the smallest possible digits at the front. With the help of additional data structures such as priority queues, we could improve both the readability and efficiency of our solution.

Here’s an example:

# Placeholder for min-heap approach in Python

Output:

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Again, this section would describe how to leverage Python’s libraries like heapq to effectively implement a min-heap approach to find the minimum possible number.

## Summary/Discussion

**Method 1: Brute Force.**Ensures the minimum number. Highly computational. Impractical for large`k`

.**Method 2: Greedy Swap.**Better performance than brute force. Not always the optimal solution.**Method 3: Optimized Bubble Swap.**Enhanced greedy approach. More efficient but can miss the absolute minimum.**Method 4: Dynamic Programming.**Improves performance for large inputs. Complex implementation.**Method 5: Min-Heap Approach.**Pythonic solution with potentially high efficiency. Depends on appropriate use of data structures.