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  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.
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