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Indistinguishable Sceneries on the Boolean Hypercube

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

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

Information

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

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