Abstract
For a variety of reasons, a number of recent works have studied the classic communication problem index, and its variant augmented-index, from a tradeoff perspective: how much communication can Alice (the player holding the n data bits) save if Bob (the player holding the index) communicates a nontrivial amount? Recently, Magniez et al. (STOC, 2010), Chakrabarti et al. (FOCS, 2010) and Jain and Nayak gave information cost tradeoffs for this problem, where the amount of communication is measured as the amount of information revealed by one player to the other. The latter two works showed that reducing Alice’s communication to sublinear requires at least a constant amount of communication from Bob.
Here, we show that the above result is just one point on a more general tradeoff curve. That is, we extend the earlier result to show that, for all b, either Bob reveals Ω(b) information to Alice, or else Alice reveals n/2O(b) information to Bob. This tradeoff lower bound is easily seen to be everywhere-tight, by virtue of an easy two-round deterministic protocol. Our lower bound applies to constant-error randomized protocols, with information measured under an “easy” distribution on inputs.
Work supported in part by NSF Grant IIS-0916565 and a McLane Family Fellowship.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ablayev, F.: Lower bounds for one-way probabilistic communication complexity and their application to space complexity. Theoretical Computer Science 175(2), 139–159 (1996)
Bar-Yossef, Z., Jayram, T.S., Krauthgamer, R., Kumar, R.: The sketching complexity of pattern matching. In: Proc. 8th International Workshop on Randomization and Approximation Techniques in Computer Science, pp. 261–272 (2004)
Bar-Yossef, Z., Jayram, T.S., Kumar, R., Sivakumar, D.: An information statistics approach to data stream and communication complexity. J. Comput. Syst. Sci. 68(4), 702–732 (2004)
Chakrabarti, A., Cormode, G., Kondapally, R., McGregor, A.: Information cost tradeoffs for augmented index and streaming language recognition. In: Proc. 51st Annual IEEE Symposium on Foundations of Computer Science, pp. 387–396 (2010)
Chakrabarti, A., Khot, S., Sun, X.: Near-optimal lower bounds on the multi-party communication complexity of set disjointness. In: Proc. 18th Annual IEEE Conference on Computational Complexity, pp. 107–117 (2003)
Chakrabarti, A., Shi, Y., Wirth, A., Yao, A.C.: Informational complexity and the direct sum problem for simultaneous message complexity. In: Proc. 42nd Annual IEEE Symposium on Foundations of Computer Science, pp. 270–278 (2001)
Clarkson, K.L., Woodruff, D.P.: Numerical linear algebra in the streaming model. In: Proc. 41st Annual ACM Symposium on the Theory of Computing, pp. 205–214 (2009)
Do Ba, K., Indyk, P., Price, E., Woodruff, D.P.: Lower bounds for sparse recovery. In: Proc. 21st Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1190–1197 (2010)
Feigenbaum, J., Kannan, S., McGregor, A., Suri, S., Zhang, J.: Graph distances in the data-stream model. SIAM J. Comput. 38(6), 1709–1727 (2008); Preliminary version in Proc. 16th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 745–754 (2005)
Gronemeier, A.: Asymptotically optimal lower bounds on the NIH-multi-party information complexity of the AND-function and disjointness. In: Proc. 26th International Symposium on Theoretical Aspects of Computer Science, pp. 505–516 (2009)
Jain, R., Nayak, A.: The space complexity of recognizing well-parenthesized expressions in the streaming model: the index function revisited. Technical Report Revision #1 to TR10-071, Electronic Colloquium on Computational Complexity (July 2010), http://eccc.hpi-web.de/
Jain, R., Radhakrishnan, J., Sen, P.: A property of quantum relative entropy with an application to privacy in quantum communication. J. ACM 56(6) (2009)
Jayram, T.S., Kumar, R., Sivakumar, D.: The one-way communication complexity of gap hamming distance. Theor. Comput. 4(1), 129–135 (2008)
Kane, D.M., Nelson, J., Woodruff, D.P.: On the exact space complexity of sketching and streaming small norms. In: Proc. 21st Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1161–1178 (2010)
Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)
Magniez, F., Mathieu, C., Nayak, A.: Recognizing well-parenthesized expressions in the streaming model. In: Proc. 41st Annual ACM Symposium on the Theory of Computing, pp. 261–270 (2010)
Miltersen, P.B., Nisan, N., Safra, S., Wigderson, A.: On data structures and asymmetric communication complexity. J. Comput. Syst. Sci. 57(1), 37–49 (1998); Preliminary version in Proc. 27th Annual ACM Symposium on the Theory of Computing, pp. 103–111 (1995)
Pǎtraşcu, M.: Unifying the landscape of cell-probe lower bounds. Manuscript (2010), http://people.csail.mit.edu/mip/papers/structures/paper.pdf
Pǎtraşcu, M., Viola, E.: Cell-probe lower bounds for succinct partial sums. In: Proc. 21st Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 117–122 (2010)
Woodruff, D.P.: Optimal space lower bounds for all frequency moments. In: Proc. 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 167–175 (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chakrabarti, A., Kondapally, R. (2011). Everywhere-Tight Information Cost Tradeoffs for Augmented Index. In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2011 2011. Lecture Notes in Computer Science, vol 6845. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22935-0_38
Download citation
DOI: https://doi.org/10.1007/978-3-642-22935-0_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22934-3
Online ISBN: 978-3-642-22935-0
eBook Packages: Computer ScienceComputer Science (R0)