Latin squares and low discrepancy allocation of two-dimensional data

Fast browsing and retrieval of geographically referenced information can require the allocation of data on different storage devices for concurrent retrieval. By dividing the two-dimensional space into tiles (essentially an array), a system can allow users to specify regions of interest using a query rectangle and then retrieving information related to tiles included in the rectangle. Suppose that there are m I/O devices. A tile is labeled by i if the data corresponding to this area is stored in the ith I/O device. A labeling is efficient if the discrepancy of the numbers of occurrences of different labels in any given rectangle is small. In the present paper, constructions are given to make this discrepancy O(logm). The constructions use Latin squares. A lower bound of Ω(logm) on the discrepancy is given for constructions of this Latin square type.