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

We show that any deterministic data-stream 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 Ω( √ n). This proves a conjecture made by Gopalan, Jayram, Krauthgamer and Kumar [10] 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 property for it.

• Type: Conference
• Year: 2007
• Where: FOCS
• Authors: Anna Gál, Parikshit Gopalan
• Keywords: Constant Factor Approximation, Deterministic Data-stream Algorithm, FOCS 2007, Longest Increasing Subsequence, Theoretical Computer Science