Intuition:

We can use a custom sorting algorithm that considers the concatenation of two numbers in different orders. Instead of sorting the entire array at once, we can compare and swap adjacent elements until the array is fully sorted according to the desired order.

Here’s how this approach works:

  • Define a custom comparison function that compares two numbers a and b by concatenating them in different orders and comparing the results.
  • Implement a custom sorting algorithm, such as Bubble Sort, Insertion Sort, or Quicksort, that uses the custom comparison function to determine the order of elements.
  • Iterate through the array and compare adjacent elements using the custom comparison function. If the elements need to be swapped to achieve the desired order, perform the swap.
  • After iterating through the array once, the array will be sorted in the desired order, with the largest possible number formed by concatenating the elements.
  • Concatenate the sorted array elements to form the largest possible number.

Solution:

Let’s implement this approach using Bubble Sort:

from typing import List

def rearrange_array(arr: List[int]) -> int:
    # Custom comparison function to compare two numbers by concatenating them in different orders
    def compare(a, b):
        return int(str(b) + str(a)) - int(str(a) + str(b))
    
    # Bubble sort implementation with the custom comparison function
    n = len(arr)
    for i in range(n):
        for j in range(0, n - i - 1):
            if compare(arr[j], arr[j + 1]) > 0:
                arr[j], arr[j + 1] = arr[j + 1], arr[j]
    
    # Concatenate the sorted array elements to form the largest possible number
    return int(''.join(map(str, arr)))

# Test cases
arr = [10, 7, 76, 415]
result = rearrange_array(arr)
print(result)  # Output should be 77641510

arr = [3, 30, 34, 5, 9]
result = rearrange_array(arr)
print(result)  # Output should be 9534330

arr = [1, 2, 3, 4, 5]
result = rearrange_array(arr)
print(result)  # Output should be 54321

Time Complexity:

  • The time complexity of this approach is O(n^2), where n is the length of the input array arr.
  • Bubble sort has a worst-case time complexity of O(n^2), and since we use a custom comparison function that takes O(m) time to compare two numbers (where m is the number of digits in the larger number), the overall time complexity is O(n^2 * m). Assuming n is significantly larger than m, it is reduced to O(n^2).
  • Note: using a more efficient sorting algorithm such as Quicksort (which takes O(n log(n))) would reduce its time complexity to O(n log(n)).

Space Complexity:

  • The space complexity is O(n) as we use additional space to store the input array and the concatenated string representation of the rearranged number.