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 Omega(radicn). This proves a conjecture made by Gopalan, Jayram, Krauthgamer and Kumar |10| who proved a matching upper bound. Our results yield asymptotically tight tower 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.