AI 中文总结
该研究提出首个基于比较的原地排序算法，以指数级高概率实现n元素数组的n lg n + O(n)次比较与O(n)次移动排序，还改进了最坏情况算法，借助新型有序集合结构缩小了最小移动量原地排序的差距。
AI 中文摘要
我们提出了首个基于比较的原地排序算法，该算法能以指数级高概率对含n个元素的数组进行排序，使用n lg n + O(n)次比较，且始终仅进行O(n)次移动。尽管仅采用线性移动且为原地操作，这一结果在附加线性项内达到了信息论下界。对于最坏情况，我们提出的算法使用n lg n + O(n lg^(t)n)次比较和O(tn)次数据移动，其中t为整数参数，满足2 ≤ t ≤ lg*n - 1，lg^(t)n表示t次迭代对数；当t>2时，该算法相比之前n lg n + O(n lg lg n)次比较、O(n)次移动的上界有所改进。我们通过随机化实现了最小移动量原地排序的终极目标，同时缩小了最坏情况下与该目标的差距，这一进展主要依赖于一种新型有序集合结构，该结构支持对n个元素进行最优的lg n + O(1)次比较的搜索操作。
英文摘要
We present the first in-place comparison-based sorting algorithm that sorts an array of $n$ elements using $n\lg n + O(n)$ comparisons with exponentially high probability and always $O(n)$ moves. This matches the information-theoretic lower bound up to an additive linear term despite making only linear moves and working in-place. For the worst-case, we present an algorithm that makes $n\lg n + O(n\lg^{(t)}n)$ comparisons and $O(tn)$ data moves, where $t$ is an integer parameter satisfying $2 \leq t \leq \lg^{*}n - 1$ and $\lg^{(t)}n$ denotes the $t$-time iterated logarithm, improving over the previous upper bound of $n\lg n + O(n\lg\lg n)$ comparisons and $O(n)$ moves when using constant $t>2$. We thus achieve the ultimate goal of minimal move in-place sorting via randomization whilst narrowing the gap to this goal in the worst-case. This advance primarily relies on a novel ordered set structure that supports searches in an optimal $\lg n + O(1)$ comparisons for $n$ elements.
Comments30 pages, 10 pages main text, 18 pages appendix
