What is the worst-case auxiliary space complexity for quick sort when it uses in-place partitioning and selects pivot randomly and without considering any stack tail-recursive optimisation?

Question

BlackTom AI · Accepted Answer

Recursive stack space counts as auxiliary space for in-place quicksort, so...

1) Relevant knowledge points and concepts
- Auxiliary space excludes the input array; it includes extra arrays, local variables, and recursion stack.
- In-place partitioning uses O(1) extra work per call (just a few indexes/temp variables).
- Recursion depth determines stack usage: each recursive call consumes constant extra space on the call stack.
- Random pivot selection affects average-case behavior but does not eliminate the possibility of worst-case recursion depth.

2) Step‑by‑step reasoning
- Each quicksort recursive call (after partitioning) recurses on two subranges. With in-place partitioning, the extra space per call is a constant c (local indices, return address, etc.).
- Worst case occurs when partitions are maximally unbalanced repeatedly (e.g., pivot always ends up as smallest or largest element). Even with random pivot selection, this worst case is still possible (just unlikely): randomness affects expected/average depth but not the theoretical worst case.
- If unbalanced splits happen N times, recursion depth becomes O(N).

3) Intermediate calculation steps
- Let c be constant stack space per recursive frame.
- Depth in worst case = N (or N − 1); total stack = c * N = O(N).
- In the balanced case depth ≈ log N ⇒ stack = c * log N = O(log N) (this is average-case, not worst-case).

4) Why the correct answer is correct
- Because the recursion stack can grow to linear size in the worst case and each frame uses constant space, the total auxiliary space is linear: O(N).

5) Why other options are incorrect
- a. O(1): false, recursion uses stack frames, so not constant in worst case.
- b. O(log N): this is the best/average-case when partitions are balanced, not the worst-case asked.
- c. O(N log N): too large — you do not allocate O(N log N) extra memory in recursive calls; stack is at most O(N).
- d. O(N^2): this is a possible time complexity in worst case, not auxiliary space for in-place partitioning.

6) Conclusion
- Therefore the worst-case auxiliary space complexity (no tail-call optimization) is O(N), matching the provided answer e. O(N).

What is the worst-case auxiliary space complexity for quick sort when it uses in-place partitioning and selects pivot randomly and without considering any stack tail-recursive optimisation?单项选择题

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