Tight Lower Bounds for the Online Labeling Problem

We consider the file maintenance problem (also called the online labeling problem) in which $n$ integer items from the set $\{1,\ldots,r\}$ are to be stored in an array of size $m \geq n$. 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 $r \leq m$ 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 moves the algorithm has to do. This problem is nontrivial for $n\le m < r$. In the case that $m = Cn$ for some $C>1$, algorithms are known that solve the problem with cost $O(n\log^2(n))$ (independent of $r$). For the case $m=n$, algorithms with cost $O(n\log^3(n))$ were given. In this paper we prove lower bounds that show that these algorithms are optimal, up to constant factors. Previously, a lower bound of $\Omega(n\log^2(n))$ was known for the restricted class of smooth algorithms [J. Zhang, Ph.D. thesis, University of Rochester, Rochester, NY].