Consider how the average of k consecutive elements can be efficiently computed using a sliding window technique to avoid recalculating the sum from scratch each time.

Explore the possibility of using a min-heap or max-heap to keep track of the smallest or largest averages, respectively, and how this can help in sorting the elements based on these averages.

Investigate the time complexity implications of your approach, particularly how the choice of data structures and algorithms affects the overall efficiency, and whether there exists an optimal solution that balances time and space complexity.