We study the problem of sorting N elements in the presence of persistent errors in comparisons: In this classical model, each comparison between two elements is wrong independently with some probability up to p, but repeating the same comparison gives always the same result. In this model, it is impossible to reliably compute a perfectly sorted permutation of the input elements. Rather, the quality of a sorting algorithm is often evaluated w.r.t. the maximum dislocation of the sequences it computes, namely, the maximum absolute difference between the position of an element in the returned sequence and the position of the same element in the perfectly sorted sequence. The best known algorithms for this problem have running time O(N²) and achieve, w.h.p., an optimal maximum dislocation of \(O(\log N)\) for constant error probability p. Note that no algorithm can achieve maximum dislocation \(o(\log N)\) w.h.p., regardless of its running time. In this work we present the first subquadratic time algorithm with optimal maximum dislocation. Our algorithm runs in \(\widetilde {O}(N^{3/2})\) time and it guarantees \(O(\log N)\) maximum dislocation with high probability for any p ≤ 1/16.

中文翻译：

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二次时间存在永久误差的最优位错

我们在比较中研究存在持久性错误的情况下对N个元素进行排序的问题：在这种经典模型中，两个元素之间的每个比较都是错误的，概率最大为p，但是重复相同的比较总是得出相同的结果。在此模型中，不可能可靠地计算输入元素的完美排序的排列。相反，排序算法的质量通常是通过最大位错来评估的所计算序列的最大差，即返回序列中元素的位置与完全排序序列中相同元素的位置之间的最大绝对差。针对该问题的最著名算法的运行时间为O（N ²），并且在恒定错误概率p的情况下，获得了最优的最大错位\（O（\ log N）\）。请注意，无论运行时间如何，任何算法都无法达到最大错位\（o（\ log N）\） whp。在这项工作中，我们提出了具有最佳最大位错的第一个二次时间算法。我们的算法以\（\ widetilde {O}（N ^ {3/2}）\）时间运行，并保证\（O（\日志N）\）最大错位以高概率为任何p ≤1/16。