The approximate sorting for big data is considered in this paper. The goal of approximate sorting for big data is to generate an approximate sorted result, but using less CPU and I/O cost. For big data, we consider the approximate sorting in I/O model. The existing metrics on permutation space are not available for external approximate sorting algorithms. Thus, we propose a new kind of metric named External metric, which ignores the errors and dislocation that happened in each I/O block.The External Spearmans footrule metric is an example of external metric for Spearmans footrule metric. Furthermore, to facilitate a better evaluation of the approximate sorted result, we propose a new metric, named as errors, which directly states the number of dislocation of the elements. Its external metric external errors is also considered in this paper. Then, according to the rate-distortion relationship endowed by these two metrics, the lower bound of these two metrics on external approximate sorting problem with t I/O operations is proved. We propose a k-pass external approximate sorting algorithm, named as EASORT, and prove that EASORT is asymptotically optimal. Finally, we consider the applications on approximate sorting results. An index for the result of our approximate sorting is proposed and analyze the single and range query on approximate sorted result using this index. Further, the sort-merge join on two relations, where one of the relations is approximate sorted or both relations are approximate sorted, are all discussed in this paper.

翻译：本文研究了大数据的近似排序问题。大数据近似排序的目标是生成近似有序的结果，同时降低CPU和I/O开销。针对大数据场景，我们在I/O模型中考虑了近似排序。现有的排列空间度量方法不适用于外部近似排序算法。因此，我们提出了一种新型度量——外部度量，该度量忽略了每个I/O块内部发生的错误与错位。外部斯皮尔曼脚距度量是斯皮尔曼脚距度量的一种外部度量实例。此外，为便于更好地评估近似排序结果，我们提出了一种名为错误量的新度量，直接表示元素的错位数量，并讨论了其外部度量形式——外部错误量。基于这两种度量所蕴含的率失真关系，我们证明了在t次I/O操作下外部近似排序问题中这两个度量的下界。我们提出了一种k趟外部近似排序算法EASORT，并证明EASORT是渐近最优的。最后，我们探讨了近似排序结果的应用。针对近似排序结果提出了一种索引，并利用该索引分析了近似排序结果上的单点查询与范围查询。本文还进一步讨论了当其中一个关系或两个关系均为近似排序时的排序-合并连接操作。