Let us analyse one by one..In each case we have to see the most expensive operation out of "findclosest , insert and delete" as T is defined as :
T = max( insert , delete , findclosest )
A) Sorted List :
For sorted list we need to find the appropriate place of insertion , so we may end up inserting at the end.. Hence T in this case = O(n)..
B) An unordered list :
Here although we can do insertion in O(1) time as there is no restriction on ordering of elements..But to delete we need to find the element hence T = deletion time = O(n)
C) An implicit heap :
Here no issue with insertion and deletion as they can be done in O(logn) time..But issue is with finding the closest pair..As we know there is no sibling relation in heap tree..So we may need to check for the entire level of the heap tree..And hence we may end up in O(n) complexity..
D) An AVL tree :
Here insertion and deletion is done in O(logn) time..To do findclosest(y) for a specific element y we can do it by finding the inorder successor and predecessor in the AVL tree along the height and hence the complexity = O(logn)..
Hence the minimum complexity is given by AVL tree..Hence D) should be the correct answer..