Which of the following statements are true?

Question

BlackTom AI · Accepted Answer

Quicksort's expected running time is O(N log N), so option (c) is correct.

1) Relevant knowledge points and concepts
- Quicksort: a divide-and-conquer sorting algorithm that partitions around a pivot, recursively sorts partitions. Partitioning each level costs O(N).
- Quickselect: selection algorithm related to quicksort used to find k-th smallest; average O(N), worst-case depends on pivot choice.
- Median-of-medians: a deterministic selection procedure that produces a “good” pivot (guaranteed to be roughly near the median) and gives worst-case linear-time selection; it does not have to return the exact median unless run specifically to compute the median.
- Time complexities: Quicksort average-case O(N log N), worst-case O(N^2) for poor pivots, best-case (balanced) O(N log N). Quickselect average O(N), worst-case O(N^2) unless median-of-medians pivot is used.

2) Step-by-step reasoning
- With a simple/random pivot selection (i.e., quicksort without median-of-medians), each partition step costs Θ(N). If pivots split the array roughly in half on average, recursion depth is Θ(log N). So total expected cost is Θ(N) work per level × Θ(log N) levels = Θ(N log N).
- A formal expected recurrence: E[T(n)] = O(n) + (1/n)∑_{k=0}^{n-1} (E[T(k)] + E[T(n-1-k)]) which solves to E[T(n)] = O(n log n).

3) Intermediate steps / calculations
- Cost per level: sum of partition work across nodes at a level = O(n).
- Number of significant levels (expected): O(log n), so total cost O(n) × O(log n) = O(n log n).

4) Why the correct answer is correct
- Option (c) states that Quicksort (using quickselect-style pivot but without median-of-medians) has average-case O(N log N). This matches standard analysis: random/typical pivot choices yield expected balanced partitions and thus average O(N log N).

5) Why the other options are incorrect
- (a) False: median-of-medians provides a guaranteed “good” pivot and can be used to deterministically select the true median if you run it for that purpose, but the statement “ensures the selected pivot is always the median for quick select” is too absolute/misleading — median-of-medians yields a pivot that is a guaranteed good splitter, not necessarily the exact median unless you run the full selection for that rank.
- (b) False: Quicksort without median-of-medians can have worst-case O(N^2) (e.g., always picking the smallest/largest element as pivot).
- (d) False: Quicksort’s best-case (balanced partitions) time is Θ(N log N), not O(N). (Quickselect can be Θ(N) for selection, but not Quicksort.)

6) Conclusion
Therefore, the true statement is (c): Quicksort with quickselect-style pivot selection without median-of-medians has an average-case running time of O(N log N), matching the provided answer.

Which of the following statements are true?多项选择题

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