# Estimating the sortedness of a data stream

## Abstract

The distance to monotonicity of a sequence is the minimum number of edit operations required to transform the sequence into an increasing order; this measure is complementary to the length of the longest increasing subsequence (LIS). We address the question of estimating these quantities in the one-pass data stream model and present the first sub-linear space algorithms for both problems. We first present O(√n)-space deterministic algorithms that approximate the distance to monotonicity and the LIS to within a factor that is arbitrarily close to 1. We also show a lower bound of Ω(n) on the space required by any randomized algorithm to compute the LIS (or alternatively the distance from monotonicity) exactly, demonstrating that approximation is necessary for sub-linear space computation; this bound improves upon the existing lower bound of Ω(√n) [LNVZ06]. Our main result is a randomized algorithm that uses only O(log2 n) space and approximates the distance to monotonicity to within a factor that is arbitrarily close to 4. In contrast, we believe that any significant reduction in the space complexity for approximating the length of the LIS is considerably hard. We conjecture that any deterministic (1 + ∈) approximation algorithm for LIS requires Ω(√n) space, and as a step towards this conjecture, prove a space lower bound of Ω(√n) for a restricted yet natural class of deterministic algorithms.