Estimating the Longest Increasing Sequence in Polylogarithmic Time

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.