The space complexity of approximating the frequency moments
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
On randomized one-round communication complexity
Computational Complexity
An Information Statistics Approach to Data Stream and Communication Complexity
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Practical Skew Handling in Parallel Joins
VLDB '92 Proceedings of the 18th International Conference on Very Large Data Bases
Counting Distinct Elements in a Data Stream
RANDOM '02 Proceedings of the 6th International Workshop on Randomization and Approximation Techniques
Information Theory Methods in Communication Complexity
CCC '02 Proceedings of the 17th IEEE Annual Conference on Computational Complexity
Tight Lower Bounds for the Distinct Elements Problem
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Asymptotic Enumeration of Graphs with a Given Upper Bound on the Maximum Degree
Combinatorics, Probability and Computing
Comparing data streams using Hamming norms (how to zero in)
VLDB '02 Proceedings of the 28th international conference on Very Large Data Bases
Buffering in query evaluation over XML streams
Proceedings of the twenty-fourth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
An improved data stream summary: the count-min sketch and its applications
Journal of Algorithms
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Foundations and Trends® in Theoretical Computer Science
Lower bounds for randomized read/write stream algorithms
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
A near-optimal algorithm for computing the entropy of a stream
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Declaring independence via the sketching of sketches
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Why go logarithmic if we can go linear?: Towards effective distinct counting of search traffic
EDBT '08 Proceedings of the 11th international conference on Extending database technology: Advances in database technology
Robust lower bounds for communication and stream computation
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Estimating PageRank on graph streams
Proceedings of the twenty-seventh ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Sketching information divergences
Machine Learning
Estimating Hybrid Frequency Moments of Data Streams
FAW '08 Proceedings of the 2nd annual international workshop on Frontiers in Algorithmics
On Estimating Frequency Moments of Data Streams
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
The average-case complexity of counting distinct elements
Proceedings of the 12th International Conference on Database Theory
A Note on Estimating Hybrid Frequency Moment of Data Streams
AAIM '09 Proceedings of the 5th International Conference on Algorithmic Aspects in Information and Management
UAI '09 Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence
A near-optimal algorithm for estimating the entropy of a stream
ACM Transactions on Algorithms (TALG)
Proceedings of the forty-second ACM symposium on Theory of computing
An optimal algorithm for the distinct elements problem
Proceedings of the twenty-ninth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Fast Manhattan sketches in data streams
Proceedings of the twenty-ninth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Information complexity: a tutorial
Proceedings of the twenty-ninth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
On the exact space complexity of sketching and streaming small norms
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Better gap-hamming lower bounds via better round elimination
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
The Computational Hardness of Estimating Edit Distance
SIAM Journal on Computing
Estimating PageRank on graph streams
Journal of the ACM (JACM)
An optimal lower bound on the communication complexity of gap-hamming-distance
Proceedings of the forty-third annual ACM symposium on Theory of computing
Fast moment estimation in data streams in optimal space
Proceedings of the forty-third annual ACM symposium on Theory of computing
Space-efficient tracking of persistent items in a massive data stream
Proceedings of the 5th ACM international conference on Distributed event-based system
Everywhere-tight information cost tradeoffs for augmented index
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
The Value of Multiple Read/Write Streams for Approximating Frequency Moments
ACM Transactions on Computation Theory (TOCT)
Optimal bounds for Johnson-Lindenstrauss transforms and streaming problems with sub-constant error
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Estimating hybrid frequency moments of data streams
Journal of Combinatorial Optimization
Tight bounds for distributed functional monitoring
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
The cost of fault tolerance in multi-party communication complexity
PODC '12 Proceedings of the 2012 ACM symposium on Principles of distributed computing
Sketching in Adversarial Environments
SIAM Journal on Computing
Synopses for Massive Data: Samples, Histograms, Wavelets, Sketches
Foundations and Trends in Databases
Optimal Bounds for Johnson-Lindenstrauss Transforms and Streaming Problems with Subconstant Error
ACM Transactions on Algorithms (TALG) - Special Issue on SODA'11
Tight lower bound for linear sketches of moments
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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We prove that any one-pass streaming algorithm which (ε, δ)-approximates the kth frequency moment Fk, for any real k ≠ 1 and any ε = Ω(1/√m), must use Ω(1/ε²) bits of space, where m is the size of the universe. This is optimal in terms of ε, resolves the open questions of Bar-Yossef et al in [3, 4], and extends the Ω(1/ε²) lower bound for F0 in [11] to much smaller ε by applying novel techniques. Along the way we lower bound the one-way communication complexity of approximating the Hamming distance and the number of bipartite graphs with minimum/maximum degree constraints.