On distance to monotonicity and longest increasing subsequence of a data stream

Authors:
Funda Ergun;Hossein Jowhari
Affiliations:
Simon Fraser University, BC, Canada;Simon Fraser University, BC, Canada
Venue:
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Year:
2008

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Abstract

In this paper we consider problems related to the sortedness of a data stream. First we investigate the problem of estimating the distance to monotonicity; given a sequence of length n, we give a deterministic (2 + ε)-approximation algorithm for estimating its distance to monotonicity in space O(1/ε2 log2 (εn)). This improves over the randomized (4 + ε)-approximation, algorithm of [3]. We then consider the problem of approximating the length of the longest increasing subsequence of an input stream of length n. We use techniques from multi-party communication complexity combined with a fooling set approach to prove that any O(1)-pass deterministic streaming algorithm that approximates the length of the longest increasing subsequence within 1 + ε requires Ω(√n) space. This proves the conjecture in [3] and matches the current upper bound.