arXiv 1804.075752018-04-20cs.DS

具有持续比较错误的最优排序

Optimal Sorting with Persistent Comparison Errors

Barbara Geissmann, Stefano Leucci, Chih-Hung Liu, Paolo Penna

备注：: 21 pages, 1 figure

摘要：

我们考虑在\emph{persistent}比较错误的情况下对$n$元素进行排序的问题。在该模型中（Braverman和Mossel，SODA'08），两个元素之间的每个比较都可能是错误的，具有一定的固定（小）概率$p$和\emph{comparisons cannot be repeated}。在这个模型中，完美的排序是不可能的，目标是最小化输出序列中每个元素的\emph{dislocation}，也就是说，它的真实排名和它的位置之间的差异。这个问题的现有下限表明，没有算法可以保证，以高概率，\emph{maximum dislocation}和\emph{total dislocation}分别优于$Ω(\log n)$和$Ω(n)$，无论其运行时间。在本文中，我们提出了第一个\emph{$O(n\log n)$-time}排序算法，保证\emph{$O(\log n)$ maximum dislocation}和\emph{$O(n)$ total dislocation}具有高概率。除了改进以前的最先进的算法-最著名的算法有运行时间$\tilde{O}(n^{3/2})$ -我们的结果表明，比较错误不使问题计算上更困难：一个序列与最好的可能的错位可以在$O(n\log n)$时间，即使没有比较错误，$Ω(n\log n)$时间是必要的，以保证这样的错位范围。为了达到这个最优结果，我们解决了两个子问题，并且各自的方法都有各自的优点，以进一步的应用。一个是如何定位在具有$O(\log n)$最大错位的几乎排序的序列中插入元素的位置，使得所得序列的错位仍然是$O(\log n)$。另一个是如何同时将$m$元素插入到几乎排序的$m$不同元素序列中，使得所得到的$2m$元素序列保持几乎排序。

英文摘要：

We consider the problem of sorting $n$ elements in the case of \emph{persistent} comparison errors. In this model (Braverman and Mossel, SODA'08), each comparison between two elements can be wrong with some fixed (small) probability $p$, and \emph{comparisons cannot be repeated}. Sorting perfectly in this model is impossible, and the objective is to minimize the \emph{dislocation} of each element in the output sequence, that is, the difference between its true rank and its position. Existing lower bounds for this problem show that no algorithm can guarantee, with high probability, \emph{maximum dislocation} and \emph{total dislocation} better than $Ω(\log n)$ and $Ω(n)$, respectively, regardless of its running time. In this paper, we present the first \emph{$O(n\log n)$-time} sorting algorithm that guarantees both \emph{$O(\log n)$ maximum dislocation} and \emph{$O(n)$ total dislocation} with high probability. Besides improving over the previous state-of-the art algorithms -- the best known algorithm had running time $\tilde{O}(n^{3/2})$ -- our result indicates that comparison errors do not make the problem computationally more difficult: a sequence with the best possible dislocation can be obtained in $O(n\log n)$ time and, even without comparison errors, $Ω(n\log n)$ time is necessary to guarantee such dislocation bounds. In order to achieve this optimal result, we solve two sub-problems, and the respective methods have their own merits for further application. One is how to locate a position in which to insert an element in an almost-sorted sequence having $O(\log n)$ maximum dislocation in such a way that the dislocation of the resulting sequence will still be $O(\log n)$. The other is how to simultaneously insert $m$ elements into an almost sorted sequence of $m$ different elements, such that the resulting sequence of $2m$ elements remains almost sorted.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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