Analysis of Approximate Sorting in I/O Model

Abstract

We consider the problem of approximate sorting in I/O model. The goal of approximate sorting in I/O model is to find out a permutation that is as close as possible to the true ordering of elements in t I/O operations. However, the quality of approximate sorting in I/O model can not be well measured by the existing metrics on permutation space. Thus, we propose a new kind of metric named External metric, which ignores the errors and dislocation that happened in each I/O block. We consider the External Spearman’s footrule metric (short for ESP) (Spearman’s footrule metric in RAM model) and a new metric external errors (short for EE) (errors in RAM model). ESP shows the block dislocation distance of each element and EE states the number of the dislocation elements in the approximate result. According to the rate-distortion relationship endowed by these two metrics, we find the lower bound of these two metrics of the permutation generated by any external approximate sorting algorithm with t I/O operations. Finally, we propose a k-pass external approximate sorting algorithm that is asymptotically optimal in I/O model.