Data structure with search, insert and delete in amortised time $O(1)$?

Problem Detail: 

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.

Asked By : A T

Answered By : A T

Looks like the $\Omega(\dfrac{\log n}{\log\log n})$ barrier has been overcome by modifying the analysis from the chronogram technique.

The new [lower] $\Omega(\log n)$ bound has been proved for similar problems in the cell-probe model [1]. From reading that article; it is my understanding that that bound applies to the list representation problem also.

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[1] Patrascu, Mihai, and Erik D. Demaine. "Logarithmic Lower Bounds in the Cell-Probe Model." SIAM J. Comput. 35, no. 4 (April 2006): 932–963. doi:10.1137/S0097539705447256.

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Question Source : http://cs.stackexchange.com/questions/1970