On the monotonicity of a data stream

Abstract

In this paper we consider problems related to the sortedness of a data stream. In the first part of this work, we investigate the problem of estimating the distance to monotonicity; given a finite stream of length n from alphabet {1,...,m}, we give a deterministic (2+є)-approximation algorithm for estimating its distance to monotonicity in space $O\left( {\tfrac{1} {{\varepsilon ^2 }}\log ^2 (\varepsilon n)} \right)$. This improves over the previous randomized (4+є)-approximation algorithm due to Gopalan, Jayram, Krauthgamer and Kumar in SODA 2007.

We then consider the problem of approximating the length of the longest increasing subsequence of the input stream. Through the analysis of a multi-party communication game, we prove that deterministic streaming algorithms that approximate the length of the longest increasing subsequence within 1+є factor require $\Omega \left( {\sqrt {\tfrac{n} {\varepsilon }} } \right)$ bits of space for any є in the range (1/n, 1/6]. This bound matches the upper bound given in the work of Gopalan et al. within a log factor. We note that, independent of our work and via a different proof strategy, Gál and Gopalan has shown an $\Omega \left( {\sqrt {\tfrac{n} {\varepsilon }} \log \left( {\tfrac{m} {{\varepsilon n}}} \right)} \right)$ lower bound for all є≥1/n including Ω(1) ranges.