Assume that the algorithms considered here sort the input sequences in ascending order. If the input is already in ascending order, which of the following are TRUE? I. Quicksort runs in $\Theta(n^2)$ time II. Bubblesort runs in $\Theta(n^2)$ time III. Mergesort runs in $\Theta(n)$ time IV. Insertion sort runs in $\Theta(n)$ time

Let us analyze the time complexity of each sorting algorithm when the input array of size $n$ is already sorted in ascending order:

• I. Quicksort: Standard Quicksort (choosing the first or last element as the pivot) produces the most unbalanced partitions (subproblems of size $0$ and $n-1$), resulting in its worst-case recurrence $T(n) = T(n-1) + \Theta(n) = \Theta(n^2)$. Hence, statement I is TRUE.

• II. Bubblesort: An optimized version of Bubblesort with an early-exit flag checks if any swap occurred and terminates in $\Theta(n)$ time on already sorted data. Hence, statement II is FALSE.

• III. Mergesort: Mergesort always divides the array into two halves and recursively merges them, running in $\Theta(n \log n)$ time in all cases (best, average, worst). Hence, statement III is FALSE.

• IV. Insertion sort: When the array is already sorted, each element requires only $1$ comparison and $0$ shifts to confirm its position, resulting in $\Theta(n)$ best-case running time. Hence, statement IV is TRUE.

Therefore, statements I and IV only are TRUE, which corresponds to option (D).