Improved Dynamic Algorithms for Longest Increasing Subsequence

TL;DR

This paper introduces the first exact dynamic algorithm for the Longest Increasing Subsequence problem with sublinear update time, and also provides improved approximate algorithms with faster update times.

Contribution

It presents a novel randomized exact dynamic LIS algorithm with sublinear update time and an improved deterministic approximate algorithm with near-constant update time.

Findings

Exact dynamic LIS algorithm with $ ilde O(n^{2/3})$ update time

Deterministic approximate LIS algorithm with $O(n^{o(1)})$ update time

Significant improvement over previous approximation algorithms

Abstract

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 $\tilde O(n^{2/3})$ 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 $\epsilon > 0$.