In the file maintenance problem, 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 locationsin the array). Each new item is stored before the next arrives. 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 or moved to a new location. The goal is to minimize the total number of such moves. The problem is non-trivial for n <= m < r.

In the case that m=Cn for some constant C>1, Itai, Konheim and Rodeh gave an algorithm that stores the items at an amortized cost of at most O(log^2(n)) per item. In the case that m = n^C for some $C>1$ their algorithm can be tuned so that the cost is O(log(n)) per item.

In this talk I'll discuss my work, in various papers, with Martin Babka, Jan Bulánek, Vladimír Čunát, and Michal Koucký, in which we establish lower bounds that match the above upper bounds.