Fast Algorithms for Bounded-Range LIS Approximation

Abstract

We introduce an improvement to additive approximation of Longest Increasing Subsequence (LIS) of a sequence with a bounded number of unique elements. In particular, for a sequence 𝑓 of length 𝑛 with 𝑟 unique elements and 𝜖 additive error paramenter, we present an algorithm that approximate the size of 𝑓’s LIS within ±𝜖𝑛 using 𝑂(𝑟𝜖⁻²) · 𝑝𝑜𝑙𝑦(log 𝜖 ⁻¹) samples and 𝑂(𝑟𝜖⁻²) · 𝑝𝑜𝑙𝑦(log 𝑟, log 𝜖 ⁻¹) runtime. Our approache introduces small adjustments to the previously known algorithm for this problem, due to [5], resulting in a polynomial runtime algorithm which uses less queries by a factor of 𝜖 ⁻¹. Similar approaches can also be applied to estimating edit distance to monotonicity in 2-dimenstional array and 𝐿₁ edit distance of a sequence within sublinear time using 𝑝𝑜𝑙𝑦(𝑟, 𝜖⁻¹) queries.