Why does linked list delete and insert operation have complexity of O(1) ? shouldn't it be of O(n)

Question

It is said that the complexity of the LinkedList remove and the add operation is of O(1). and in case of ArrayList it is of O(n).

Calculation for ArrayList of size "M" : if i want to remove the element at Nth position then i can directly go to the Nth position using index in one go (i don't have to traverse till Nth index) and then i can remove the element, till this point the complexity is O(1) then i will have to shift the rest of the elements(M-N shifts) so my complexity will be linear i.e. O(M-N+1). and hence deletion or insertion at the last will give me the best performence( as N ~ M) and deletion or insertion at the start will be worst (as N ~ 1).

Now the LisnkedList of size "M" : as we can not directly reach the Nth element in the LinkedList, to access the Nth element we have to traverse N elements, so the search in the LinkedList is costlier then the ArrayList...but Remove and the add operations are said to be of O(1) in case of LinkedList as, in LinkedList the Shift is not involved, but there is traverse operation involved rigth ? so the complexity should be of order O(n) where Worst performence will be at the tail node and best performence will be at the head node.

Could anyone please explain me why don't we consider the traverse cost while calculating the complexity of LinkedList remove operation ?

So i wants to understand how it works in java.util package. and if i want to implemet the same in C or C++ how would i achieve the O(1) for random deletion and insertion in LinkedList ?

Answer 1

Remove and the add operations are said to be of O(1) in case of LinkedList as, in LinkedList the shift is not involved, but there is traverse operation involved right?

Adding to either end of a linked list does not require a traversal, as long as you keep a reference to both ends of the list. This is what Java does for its add and addFirst/addLast methods.

Same goes for parameterless remove and removeFirst/removeLast methods - they operate on list ends.

remove(int) and remove(Object) operations, on the other hand, are not O(1). They requires traversal, so you correctly identified their costs as O(n).