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

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 + )-approximation algorithm for estimating its distance to monotonicity in space O( 1 2 log2 ( n)). This improves over the randomized (4 + )-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+ requires ( n) space. This proves the conjecture in [3] and matches the current upper bound.

• Type: Conference
• Year: 2008
• Where: SODA
• Authors: Funda Ergün, Hossein Jowhari
• Keywords: )-approximation Algorithm, Algorithms, Deterministic Streaming Algorithm, Longest Increasing Subsequence, SODA 2008