Find the kth largest element in an array

Question presented to candidate: "Given an unsorted list of integers and an integer k, return the kth largest element — the kth in sorted descending order, not the kth distinct. Please solve it in Python, and tell me the trade-offs between the approaches you know."

What a strong answer should cover:

• Sorting first: sorted(nums, reverse=True)[k - 1], O(n log n).
• A min-heap of size k with heapq: push, pop when the size exceeds k, answer is heap[0]; O(n log k) time, O(k) space; why a min-heap.
• Quickselect at index n - k: O(n) average, O(n²) worst case, in place on a copy.
• When to choose which: streams and small k favour the heap; one-off in-memory queries favour Quickselect.
• Python specifics: heapq is a min-heap only (negate values for a max-heap), heapq.heappushpop, heapq.nlargest(k, nums)[-1], and that the heap list is not sorted.

Clarifying questions expected:

• "Do duplicates count separately?"
• "Is the data a stream or a list in memory?"

Code / implementation expected: Yes — at least the heapq solution in Python; ideally also Quickselect.