Non-interactive correlation distillation, inhomogeneous Markov chains, and the reverse Bonami-Beckner inequality
- 126 Downloads
In this paper we studynon-interactive correlation distillation (NICD), a generalization of noise sensitivity previously considered in [5, 31, 39]. We extend the model toNICD on trees. In this model there is a fixed undirected tree with players at some of the nodes. One node is given a uniformly random string and this string is distributed throughout the network, with the edges of the tree acting as independent binary symmetric channels. The goal of the players is to agree on a shared random bit without communicating.
• In the case of ak-leaf star graph (the model considered in ), we resolve the open question of whether the success probability must go to zero ask » ∞. We show that this is indeed the case and provide matching upper and lower bounds on the asymptotically optimal rate (a slowly-decaying polynomial).
• In the case of thek-vertex path graph, we show that it is always optimal for all players to use the same 1-bit function.
• In the general case we show that all players should use monotone functions. We also show, somewhat surprisingly, that for certain trees it is better if not all players use the same function.
Our techniques include the use of thereverse Bonami-Beckner inequality. Although the usual Bonami-Beckner has been frequently used before, its reverse counterpart seems not to be well known. To demonstrate its strength, we use it to prove a new isoperimetric inequality for the discrete cube and a new result on the mixing of short random walks on the cube. Another tool that we need is a tight bound on the probability that a Markov chain stays inside certain sets; we prove a new theorem generalizing and strengthening previous such bounds [2, 3, 6]. On the probabilistic side, we use the “reflection principle” and the FKG and related inequalities in order to study the problem on general trees.
Unable to display preview. Download preview PDF.
- M. Ajtai, J. Komolós and E. Szemerédi,Deterministic simulation in LOGSPACE, inProceedings of the 19th Annual ACM Symposium on Theory of Computing, ACM, New York, 1987, pp. 132–140.Google Scholar
- N. Alon, U. Feige, A. Wigderson and D. Zuckerman,Derandomized graph products, Computational Complexity, Birkhäuser, Basel, 1995.Google Scholar
- N. Alon, U. Maurer and A. Wigderson, Unpublished results, 1991.Google Scholar
- K. Amano and A. Maruoka,On learning monotone Boolean functions under the uniform distribution, Lecture Notes in Computer Science2533, Springer, New York, 2002, pp. 57–68.Google Scholar
- M. Ben-Or and N. Linial,Collective coin flipping, inRandomness and Computation (S. Micali, ed.), Academic Press, New York, 1990.Google Scholar
- N. Bshouty, J. Jackson and C. Tamon,Uniform-distribution attribute noise learnability, inProceedings of the Eighth Annual Conference on Computational Learning Theory, (COLT 1995), Santa Cruz, California, USA, ACM, 1995.Google Scholar
- I. Dinur, V. Guruswami and S. Khot,Vertex cover on k-uniform hypergraphs is hard to approximate within factor (k-3-∈), ECCC Technical Report TRO2-027, 2002.Google Scholar
- I. Dinur and S. Safra,The importance of being biased, inProceedings of the 34th Annual ACM Symposium on the Theory of Computing, ACM, New York, 2002, pp. 33–42.Google Scholar
- G. Hardy, J. Littlewood and G. Póyla,Inequalities, 2nd edn., Cambridge University Press, 1952.Google Scholar
- J. Kahn, G. Kalai and N. LinialThe influence of variables on boolean functions, inProceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1988, pp. 68–80.Google Scholar
- S. Khot,On the power of unique 2-prover 1-round games, inProceedings of the 34th Annual ACM Symposium on the Theory of Computing, ACM, New York, 2002, pp. 767–775.Google Scholar
- R. O'Donnell,Hardness amplification within NP, inProceedings of the 34th Annual ACM Symposium on the Theory of Computing, ACM, New York, 2002, pp. 715–760.Google Scholar
- R. O'Donnell,Computational applications of noise sensitivity, PhD thesis, Massachusetts Institute of Technology, 2003.Google Scholar
- R. O'Donnell and R. Servedio,Learning monotone decision trees, Manuscript, 2004.Google Scholar
- M. Talagrand,On Russo's approximate 0–1 law, Annals of Probability22 (1994), 1476–1387.Google Scholar
- K. Yang,On the (im)possibility of non-interactive correlation distillation, inLATIN 2004: Theoretical Informatics, 6th Latin American Symposium, Buenos Aires, Argentina, April 5–8, 2004, Proceedings (M. Farach-Colton, ed.), Lecture Notes in Computer Science 2976, Springer, Berlin, 2004.Google Scholar