Approximating the Longest Increasing Sequence and Distance from Sortedness in a Data Stream

We revisit the well-studied problem of estimating the sortedness of a data stream. We study the complementary problems of estimating the edit distance from sortedness (Ulam distance) and estimating the length of the longest increasing sequence (LIS). We present the first sub-linear space algorithms for these problems in the data stream model. • We give a O(log n) space, one-pass randomized algorithm that gives a (4 + ) approximation to the Ulam distance. • We O( √ n) space deterministic (1 + ) one-pass approximation algorithms for estimating the the length of the LIS and the Ulam distance. • We show a tight lower bound of Ω(n) on the space required by any randomized algorithm to compute these quantities exactly. This improves an Ω( √ n) lower bound due to Vee et al. and shows that approximation is essential to get space-efficient algorithms. • We conjecture a space lower bound of Ω( √ n) on any deterministic algorithm approximating the LIS. We are able to show such a bound for a restricted class of algorithms, which nevertheless captures all the algorithms described above. Our algorithms and lower bounds use techniques from communication complexity and property testing. ∗Work done in part while the author was at IBM Almaden