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Indistinguishable Sceneries on the Boolean Hypercube

Part of: Graph theory Theory of data Discrete mathematics in relation to computer science

Published online by Cambridge University Press:  05 June 2018

RENAN GROSS  and
URI GRUPEL
RENAN GROSS
Affiliation:
Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Herzl Street, Rehovot 7610001, Israel (e-mail: renan.gross@weizmann.ac.il, uri.grupel@weizmann.ac.il)
URI GRUPEL
Affiliation:
Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Herzl Street, Rehovot 7610001, Israel (e-mail: renan.gross@weizmann.ac.il, uri.grupel@weizmann.ac.il)
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Abstract

We show that the scenery reconstruction problem on the Boolean hypercube is in general impossible. This is done by using locally biased functions, in which every vertex has a constant fraction of neighbours coloured by 1, and locally stable functions, in which every vertex has a constant fraction of neighbours coloured by its own colour. Our methods are constructive, and also give super-polynomial lower bounds on the number of locally biased and locally stable functions. We further show similar results for ℤn and other graphs, and offer several follow-up questions.

MSC classification

Primary: 05C60: Isomorphism problems (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)
Secondary: 68P30: Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) 68R05: Combinatorics
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 1 , January 2019 , pp. 46 - 60
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

†

Supported by the European Research Council (ERC).

References

[1] Benjamini, I. and Kesten, H. (1996) Distinguishing sceneries by observing the scenery along a random walk path. J. Anal. Math. 69 97–135.Google Scholar
[2] Finucane, H., Tamuz, O. and Yaari, Y. (2014) Scenery reconstruction on finite abelian groups. Stochastic Process. Appl. 124 2754–2770.Google Scholar
[3] Garban, C. and Steif, J. E. (2015) Noise Sensitivity of Boolean Functions and Percolation, Institute of Mathematical Statistics Textbooks, Cambridge University Press.Google Scholar
[4] Hardy, G. H. and Ramanujan, S. (1918) Asymptotic formulæ in combinatory analysis. Proc. London Math. Soc. s2-17 75–115.Google Scholar
[5] Krotov, D. S. and Avgustinovich, S. V. (2008) On the number of 1-perfect binary codes: A lower bound. IEEE Trans. Inform. Theor. 54 1760–1765.Google Scholar
[6] Lindenstrauss, E. (1999) Indistinguishable sceneries. Random Struct. Alg. 14 71–86.Google Scholar
[7] van Lint, J. H. (1975) A survey of perfect codes. Rocky Mountain J. Math. 5 199–224.Google Scholar
[8] van Lint, J. H. (1998) Introduction to Coding Theory, third edition, Springer.Google Scholar
[9] Matzinger, H. and Rolles, S. W. (2003) Reconstructing a piece of scenery with polynomially many observations. Stochastic Process. Appl. 107 289–300.Google Scholar
[10] O'Donnell, R. (2014) Analysis of Boolean Functions, Cambridge University Press.Google Scholar