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Indistinguishable Sceneries on the Boolean Hypercube

Published online by Cambridge University Press:  05 June 2018

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)
Corresponding

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.

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Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

Supported by the European Research Council (ERC).

References

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