1 research outputs found
Tight lower bounds for online labeling problem
We consider the file maintenance problem (also called the online labeling
problem) in which n integer items from the set {1,...,r} are to be stored in an
array of size m >= n. The items are presented sequentially in an arbitrary
order, and must be stored in the array in sorted order (but not necessarily in
consecutive locations in the array). Each new item must be stored in the array
before the next item is received. If r<=m then we can simply store item j in
location j but if r>m then we may have to shift the location of stored items to
make space for a newly arrived item. The algorithm is charged each time an item
is stored in the array, or moved to a new location. The goal is to minimize the
total number of such moves done by the algorithm. This problem is non-trivial
when n=<m<r.
In the case that m=Cn for some C>1, algorithms for this problem with cost
O(log(n)^2) per item have been given [IKR81, Wil92, BCD+02]. When m=n,
algorithms with cost O(log(n)^3) per item were given [Zha93, BS07]. In this
paper we prove lower bounds that show that these algorithms are optimal, up to
constant factors. Previously, the only lower bound known for this range of
parameters was a lower bound of \Omega(log(n)^2) for the restricted class of
smooth algorithms [DSZ05a, Zha93].
We also provide an algorithm for the sparse case: If the number of items is
polylogarithmic in the array size then the problem can be solved in amortized
constant time per item.Comment: 24 pages, 1 figur