The Highest Lower Bound on the number of Comparisons in the worst case for comparison-based sorting order of : 

The correct answer is n log n.

Key Points

• In comparison-based sorting algorithms, the number of comparisons required to sort a list of n elements in the worst case is at least proportional to n log n.
• This result is derived from the fact that there are n! possible permutations of n elements, and each comparison can at most halve the number of possibilities.
• Therefore, to distinguish between all n! permutations, we need at least log(n!) comparisons.
• Using Stirling's approximation, log(n!) is Θ(n log n), which provides the lower bound for comparison-based sorting algorithms.
• This lower bound applies to all comparison-based sorting algorithms, including well-known ones like Merge Sort, Heap Sort, and Quick Sort (on average).

Additional Information

• Non-comparison-based sorting algorithms, such as Counting Sort, Radix Sort, and Bucket Sort, can achieve better performance under specific conditions, bypassing the n log n lower bound.
• These algorithms rely on additional assumptions about the input data, such as the range of the input values.
• However, for general-purpose sorting where no assumptions can be made about the input data, comparison-based sorting algorithms are the most widely applicable.
• Understanding the lower bound helps in analyzing and comparing the efficiency of different sorting algorithms.
• The n log n lower bound ensures that no comparison-based sorting algorithm can have a worst-case performance better than this bound.