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If you want to maintain a sorted list which you will frequently modify (i.e. a structure which, in addition to being sorted, allows duplicates and whose elements can be efficiently referenced by index), then use an ArrayList but when you need to insert an element, always use Collections.binarySearch() to determine the index at which you add a given element. The latter method tells you the index you need to insert at to keep your list in sorted order.
TreeSet would not work because they do not allow duplicates plus they do not provide method to fetch element at specific position. PriorityQueue would not work because it does not allow fetching elements at specific position which is a basic requirement for a list. I think you need to implement your own algorithm to maintain a sorted list in Java with O(logn) insert time, unless you do not need duplicates. Maybe a solution could be using a TreeMap where the key is a subclass of the item overriding the equals method so that duplicates are allowed.