We study dynamic algorithms for the longest increasing subsequence (\textsf{LIS}) problem. A dynamic \textsf{LIS} algorithm maintains a sequence subject to operations of the following form arriving one by one: (i) insert an element, (ii) delete an element, or (iii) substitute an element for another. After performing each operation, the algorithm must report the length of the longest increasing subsequence of the current sequence. Our main contribution is the first exact dynamic \textsf{LIS} algorithm with sublinear update time. More precisely, we present a randomized algorithm that performs each operation in time $\tilde O(n^{4/5})$ and after each update, reports the answer to the \textsf{LIS} problem correctly with high probability. We use several novel techniques and observations for this algorithm that may find their applications in future work. In the second part of the paper, we study approximate dynamic \textsf{LIS} algorithms, which are allowed to underestimate the solution size within a bounded multiplicative factor. In this setting, we give a deterministic algorithm with update time $O(n^{o(1)})$ and approximation factor $1-o(1)$. This result substantially improves upon the previous work of Mitzenmacher and Seddighin (STOC'20) that presents an $\Omega(\epsilon ^{O(1/\epsilon)})$-approximation algorithm with update time $\tilde O(n^\epsilon)$ for any constant $\epsilon > 0$.

中文翻译：

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最长递增子序列的改进动态算法

我们研究了最长的递增子序列（\ textsf {LIS}）问题的动态算法。动态\ textsf {LIS}算法维护序列，该序列遵循以下形式的操作：一个（i）插入一个元素，（ii）删除一个元素，或（iii）将一个元素替换为另一个。执行完每个操作后，算法必须报告当前序列中最长的递增子序列的长度。我们的主要贡献是第一个具有亚线性更新时间的精确动态\ textsf {LIS}算法。更精确地讲，我们提出了一种随机算法，该算法在时间$ \ tilde O（n ^ {4/5}）$中执行每个操作，并在每次更新后以很高的概率正确报告\ textsf {LIS}问题的答案。我们对该算法使用了几种新颖的技术和观察，它们可能会在未来的工作中找到应用。在本文的第二部分中，我们研究了近似动态\ textsf {LIS}算法，该算法可以低估有界乘法因子内的解大小。在这种设置下，我们给出了一种确定性算法，其中更新时间为$ O（n ^ {o（1）}）$，逼近因子为$ 1-o（1）$。该结果大大改进了Mitzenmacher和Seddighin（STOC'20）的先前工作，该工作提出了更新时间为$ \ tilde O（n的$ \ Omega（\ epsilon ^ {O（1 / \ epsilon）}）$近似算法。对于任何常量$ \ epsilon> 0 $，^ \ epsilon）$。