Is there a data structure to maintain an ordered list that supports the following operations in $O(1)$ amortized time?
GetElement(k): Return the $k$th element of the list.
InsertAfter(x,y): Insert the new element y into the list immediately after x.
Delete(x): Remove x from the list.
For the last two operations, you can assume that x is given as a pointer directly into the data structure; InsertElement returns the corresponding pointer for y. InsertAfter(NULL, y) inserts y at the beginning of the list.
For example, starting with an empty data structure, the following operations update the ordered list as shown below:
- InsertAfter(NULL, a) $\implies$ [a]
- InsertAfter(NULL, b) $\implies$ [b, a]
- InsertAfter(b, c) $\implies$ [b, c, a]
- InsertAfter(a, d) $\implies$ [b, c, a, d]
- Delete(c) $\implies$ [b, a, d]
After these five updates, GetElement(2) should return d, and GetElement(3) should return an error.