On Preserving Proximity in Extendible Arrays

A. L. Rosenberg [4] has shown that fully extendible array storage schemes cannot preserve proximity of array positions in any global sense but only in a local sense. The purpose of this note is to consider two problems suggested in [4]. Our first result shows that, for any k array positions of an extendible array, there is a storage scheme which stores array elements that are “close to” the given k positions optimally “close” to each other. The “local diameter of preservation” of a storage scheme is essentially the cardinality of the smallest set of contiguous locations assigned to array positions in a “neighborhood” of a given array position. Our second result shows the nonexistence of “almost everywhere” polynomial bounds of a certain type for local diameters of preservation.