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 $Omega(dfrac{log n}{loglog n})$ 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. [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
