在线列表标签：打破$\log^2n$障碍
Online List Labeling: Breaking the $\log^2n$ Barrier
摘要：
在线列表标签问题是一个算法的原始与大量的文献的上限，下限，和应用。目标是在$m$插槽数组中存储一组动态变化的$n$项，同时保持项以排序顺序出现的不变式，并最小化重新标记成本，定义为每次插入/删除移动的项的数量。对于线性制度，其中$m = (1 + Θ(1)) n$，自1981年以来，$O(\log^2 n)$对重新标记成本的上限已为人所知。$Ω(\log^2 n)$的下界对于确定性算法和所谓的平滑算法是已知的，但是最好的一般下界仍然是$Ω(\log n)$。该领域的中心问题是$O(\log^2 n)$是否对所有算法都是最优的。在本文中，我们给出了一个随机化的数据结构，达到预期的重新标记成本$O(\log^{3/2} n)$每操作。更一般地，如果$m = (1 + \varepsilon) n$用于$\varepsilon = O(1)$，则预期的重新标记成本变为$O(\varepsilon^{-1} \log^{3/2} n)$。我们的解决方案是历史无关的，这意味着数据结构的状态与插入/删除项的顺序无关。对于历史无关的数据结构，我们还证明了一个匹配的下界：对于所有$ε$之间的$1 / n^{1/3}$和一些足够小的正常数，最佳的预期成本为历史无关的列表标签的解决方案是$Θ(\varepsilon^{-1}\log^{3/2} n)$。
英文摘要：
The online list labeling problem is an algorithmic primitive with a large literature of upper bounds, lower bounds, and applications. The goal is to store a dynamically-changing set of $n$ items in an array of $m$ slots, while maintaining the invariant that the items appear in sorted order, and while minimizing the relabeling cost, defined to be the number of items that are moved per insertion/deletion. For the linear regime, where $m = (1 + Θ(1)) n$, an upper bound of $O(\log^2 n)$ on the relabeling cost has been known since 1981. A lower bound of $Ω(\log^2 n)$ is known for deterministic algorithms and for so-called smooth algorithms, but the best general lower bound remains $Ω(\log n)$. The central open question in the field is whether $O(\log^2 n)$ is optimal for all algorithms. In this paper, we give a randomized data structure that achieves an expected relabeling cost of $O(\log^{3/2} n)$ per operation. More generally, if $m = (1 + \varepsilon) n$ for $\varepsilon = O(1)$, the expected relabeling cost becomes $O(\varepsilon^{-1} \log^{3/2} n)$. Our solution is history independent, meaning that the state of the data structure is independent of the order in which items are inserted/deleted. For history-independent data structures, we also prove a matching lower bound: for all $ε$ between $1 / n^{1/3}$ and some sufficiently small positive constant, the optimal expected cost for history-independent list-labeling solutions is $Θ(\varepsilon^{-1}\log^{3/2} n)$.
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。
