Storing a Sparse Table with 0(1) Worst Case Access Time
Journal of the ACM (JACM)
Private vs. common random bits in communication complexity
Information Processing Letters
Lower bounds for union-split-find related problems on random access machines
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
Approximate nearest neighbors: towards removing the curse of dimensionality
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
On data structures and asymmetric communication complexity
Journal of Computer and System Sciences
Optimal bounds for the predecessor problem
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Lower bounds for high dimensional nearest neighbor search and related problems
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Journal of the ACM (JACM)
Tighter bounds for nearest neighbor search and related problems in the cell probe model
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Introduction to Modern Information Retrieval
Introduction to Modern Information Retrieval
Efficient Search for Approximate Nearest Neighbor in High Dimensional Spaces
SIAM Journal on Computing
An Information Statistics Approach to Data Stream and Communication Complexity
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Informational Complexity and the Direct Sum Problem for Simultaneous Message Complexity
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
A Replacement for Voronoi Diagrams of Near Linear Size
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
A strong lower bound for approximate nearest neighbor searching
Information Processing Letters
An Optimal Randomised Cell Probe Lower Bound for Approximate Nearest Neighbour Searching
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Cell-probe lower bounds for the partial match problem
Journal of Computer and System Sciences - Special issue: STOC 2003
On the Optimality of the Dimensionality Reduction Method
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
The Communication Complexity of Correlation
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
Probabilistic computations: Toward a unified measure of complexity
SFCS '77 Proceedings of the 18th Annual Symposium on Foundations of Computer Science
Higher Lower Bounds for Near-Neighbor and Further Rich Problems
SIAM Journal on Computing
A direct sum theorem in communication complexity via message compression
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
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We consider the approximate nearest neighbor search problem on the Hamming cube $\{0,1\}^d$. We show that a randomized cell probe algorithm that uses polynomial storage and word size $d^{O(1)}$ requires a worst case query time of $\Omega({\rm log}\,{\rm log}\,d/{\rm log}\,{\rm log}\,{\rm log}\,d)$. The approximation factor may be as loose as $2^{{\rm log}^{1-\eta}d}$ for any fixed $\eta0$. Our result fills a major gap in the study of this problem since all earlier lower bounds either did not allow randomization [A. Chakrabarti et al., A lower bound on the complexity of approximate nearest-neighbor searching on the Hamming cube, in Discrete and Computational Geometry, Springer, Berlin, 2003, pp. 313-328; D. Liu, Inform. Process. Lett., 92 (2004), pp. 23-29] or did not allow approximation [A. Borodin, R. Ostrovsky, and Y. Rabani, Proceedings of the 31st Annual ACM Symposium on Theory of Computing, 1999, pp. 312-321; O. Barkol and Y. Rabani, Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, 2000, pp. 388-396; T. S. Jayram et al., J. Comput. System Sci., 69 (2004), pp. 435-447]. We also give a cell probe algorithm that proves that our lower bound is optimal. Our proof uses a lower bound on the round complexity of the related communication problem. We show, additionally, that considerations of bit complexity alone cannot prove any nontrivial cell probe lower bound for the problem. This shows that the “richness technique” [P. B. Miltersen et al., J. Comput. System Sci., 57 (1998), pp. 37-49] used in a lot of recent research around this problem would not have helped here. Our proof is based on information theoretic techniques for communication complexity, a theme that has been prominent in recent research [A. Chakrabarti et al., Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science, 2001, pp. 270-278; Z. Bar-Yossef et al., Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002, pp. 209-218; P. Sen, Proceedings of the 18th Annual IEEE Conference on Computational Complexity, 2003, pp. 73-83; R. Jain, J. Radhakrishnan, and P. Sen, Proceedings of the 30th International Colloquium on Automata, Languages and Programming, 2003, pp. 300-315].