Analysis of Approximate sorting in I/O model

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 \textit{External Spearman's footrule metric} (short for \textit{ESP}) (\textit{Spearman's footrule metric} in RAM model) and a new metric \textit{external errors} (short for \textit{EE}) (\textit{errors} in RAM model). \textit{ESP} shows the block dislocation distance of each element and \textit{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.