Estimating the longest increasing sequence in polylogarithmic time

TL;DR

This paper introduces a novel randomized algorithm that estimates the length of the longest increasing subsequence in an array within a small additive error in polylogarithmic time, significantly improving over previous methods.

Contribution

The paper presents the first polylogarithmic time algorithm for approximating LIS length with arbitrary precision, surpassing prior algorithms limited to coarse approximations.

Findings

Achieves polylogarithmic time complexity for LIS estimation.

Provides additive and multiplicative approximation guarantees.

Improves approximation quality over previous algorithms.

Abstract

Finding the length of the longest increasing subsequence (LIS) is a classic algorithmic problem. Let $n$ denote the size of the array. Simple $O(n\log n)$ algorithms are known for this problem. We develop a polylogarithmic time randomized algorithm that for any constant $\delta > 0$, estimates the length of the LIS of an array to within an additive error of $\delta n$. More precisely, the running time of the algorithm is $(\log n)^c (1/\delta)^{O(1/\delta)}$ where the exponent $c$ is independent of $\delta$. Previously, the best known polylogarithmic time algorithms could only achieve an additive $n/2$ approximation. With a suitable choice of parameters, our algorithm also gives, for any fixed $\tau>0$, a multiplicative $(1+\tau)$-approximation to the distance to monotonicity $\varepsilon_f$ (the fraction of entries not in the LIS), whose running time is polynomial in $\log(n)$ and $1/varepsilon_f$. The best previously known algorithm could only guarantee an approximation within a factor (arbitrarily close to) 2.