On distance to monotonicity and longest increasing subsequence of a data stream
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Revisiting the Direct Sum Theorem and Space Lower Bounds in Random Order Streams
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Streaming algorithms with one-sided estimation
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Lower bounds for number-in-hand multiparty communication complexity, made easy
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
A note on randomized streaming space bounds for the longest increasing subsequence problem
Information Processing Letters
Edit distance to monotonicity in sliding windows
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
Tight bounds for distributed functional monitoring
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Space-bounded communication complexity
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Streaming computations with a loquacious prover
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Communication steps for parallel query processing
Proceedings of the 32nd symposium on Principles of database systems
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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 \left( {\sqrt n } \right). 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 type property for it.