Fully Dynamic Approximation of LIS in Polylogarithmic Time

TL;DR

This paper presents a new dynamic algorithm that maintains a near-optimal approximation of the Longest Increasing Subsequence (LIS) in polylogarithmic worst-case time, significantly improving previous results and impacting related problems like Erd ext{"o}s-Szekeres partitioning.

Contribution

The authors introduce a novel LIS sparsification technique and develop an algorithm that maintains a (1+ε)-approximate LIS with polylogarithmic worst-case update time, surpassing prior exponential-time approximations.

Findings

Achieved (1+ε)-approximation of LIS with polylogarithmic update time.

Improved Erd ext{"o}s-Szekeres partitioning to near-linear time.

Provided a new approach based on LIS sparsification, different from previous grid packing methods.

Abstract

We revisit the problem of maintaining the longest increasing subsequence (LIS) of an array under (i) inserting an element, and (ii) deleting an element of an array. In a recent breakthrough, Mitzenmacher and Seddighin [STOC 2020] designed an algorithm that maintains an $\mathcal{O}((1/\epsilon)^{\mathcal{O}(1/\epsilon)})$-approximation of LIS under both operations with worst-case update time $\mathcal{\tilde O}(n^{\epsilon})$, for any constant $\epsilon>0$. We exponentially improve on their result by designing an algorithm that maintains an $(1+\epsilon)$-approximation of LIS under both operations with worst-case update time $\mathcal{\tilde O}(\epsilon^{-5})$. Instead of working with the grid packing technique introduced by Mitzenmacher and Seddighin, we take a different approach building on a new tool that might be of independent interest: LIS sparsification. A particularly interesting consequence of our result is an improved solution for the so-called Erd\H{o}s-Szekeres partitioning, in which we seek a partition of a given permutation of $\{1,2,\ldots,n\}$ into $\mathcal{O}(\sqrt{n})$ monotone subsequences. This problem has been repeatedly stated as one of the natural examples in which we see a large gap between the decision-tree complexity and algorithmic complexity. The result of Mitzenmacher and Seddighin implies an $\mathcal{O}(n^{1+\epsilon})$ time solution for this problem, for any $\epsilon>0$. Our algorithm (in fact, its simpler decremental version) further improves this to $\mathcal{\tilde O}(n)$.