Lower Bounds on Streaming Algorithms for Approximating the Length of the Longest Increasing Subsequence

Authors:
Anna Gál;Parikshit Gopalan
Affiliations:
panni@cs.utexas.edu;parik@microsoft.com
Venue:
SIAM Journal on Computing
Year:
2010

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Abstract

We show that any deterministic streaming algorithm that makes a constant number of passes over the input and gives a constant factor approximation of the length of the longest increasing subsequence in a sequence of length $n$ must use space $\Omega(\sqrt{n})$. This proves a conjecture made by Gopalan et al. [Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, 2007, pp. 318-327] who proved a matching upper bound. Our results yield asymptotically tight lower bounds for all approximation factors, thus resolving the main open problem from their paper. Our proof is based on analyzing a related communication problem and proving a direct sum type property for it.