Time complexity of deleting random element from heap

#1

Time complexity is logn, but here we also have to find that key and that will take maybe (n), so what about this?

#2

that is true.
to delete a random element, we need to first search the element which is O(n) complexity. so delete operation is O(n).

But in this question we presume that index of element is given, which skips searching part. thus reduces time complexity to O(logn).

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#3

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