Fully Dynamic Approximation of LIS in Polylogarithmic Time

11/19/2020 ∙ by Paweł Gawrychowski, et al. ∙ 0

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 𝒪((1/ϵ)^𝒪(1/ϵ))-approximation of LIS under both operations with worst-case update time Õ(n^ϵ), for any constant ϵ>0. We exponentially improve on their result by designing an algorithm that maintains an (1+ϵ)-approximation of LIS under both operations with worst-case update time Õ(ϵ^-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 Erdős-Szekeres partitioning, in which we seek a partition of a given permutation of {1,2,…,n} into 𝒪(√(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 𝒪(n^1+ϵ) time solution for this problem, for any ϵ>0. Our algorithm (in fact, its simpler decremental version) further improves this to Õ(n).

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