Final answer:
The running time of Heapsort on an array with all equal key values remains Θ(nlogn). While building the heap, comparisons occur with no swaps, leading to an overall efficient running time despite all elements being equal.
Step-by-step explanation:
Running Time of Heapsort with Equal Key Values
When discussing the running time of Heapsort on an array where all key values are equal, it is crucial to understand the intricacies of the algorithm. Heapsort utilizes a binary heap structure and proceeds in two main phases: building the heap and then repeatedly extracting the maximum element to achieve a sorted array. In the case where all elements are equal, building the heap would still require comparing elements, but every comparison would result in no swaps since the values are identical.
In this specific scenario, the algorithm still runs through its motions but with fewer swaps. As a result, the running time of Heapsort remains relatively efficient since the most costly operations typically involve data movement. Hence, even in the presence of equal key values, Heapsort's running time is Θ(n − logn), where 'n' is the number of elements in the array. The complexity does not degrade to quadratic time as it might with certain other sorting algorithms in their worst-case scenarios.
Therefore, the correct answer to the question is e. Θ(nlogn).