Which one of the statement given in the options is true about the Shell sort?

Shell sort's time complexity depends on the gap sequence used. With an optimal gap sequence (like Knuth's: 3k+1), the average case time complexity is O(n^4/3). The worst case is O(n²) when using a poor gap sequence, and the best case is O(n) when the array is already sorted. Option D correctly identifies the average case complexity.

Step-by-step Derivation:
Shell sort analysis:

• Best case: O(n) — occurs when the array is already sorted or nearly sorted, requiring only linear comparisons and shifts.
• Average case: O(n^4/3) — with optimal gap sequences (e.g., Knuth's sequence 1, 4, 13, 40...), empirically shown to be O(n^4/3).
• Worst case: O(n²) — occurs with suboptimal gap sequences or worst-case input arrangements.

The gap sequence significantly affects performance. Knuth's sequence (3k+1) and other optimized sequences achieve the O(n^4/3) average case. Options A and C claim impossible/wrong complexities. Option B is incorrect because best case is O(n), not O(n^4/3).