TY - JOUR
T1 - Indistinguishable Sceneries on the Boolean Hypercube
AU - Gross, Renan
AU - Grupel, Uri
N1 - Publisher Copyright:
© Cambridge University Press 2018.
PY - 2019/1/1
Y1 - 2019/1/1
N2 - 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 ℤ n and other graphs, and offer several follow-up questions.
AB - 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 ℤ n and other graphs, and offer several follow-up questions.
UR - http://www.scopus.com/inward/record.url?scp=85047981580&partnerID=8YFLogxK
U2 - 10.1017/S0963548318000305
DO - 10.1017/S0963548318000305
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AN - SCOPUS:85047981580
SN - 0963-5483
VL - 28
SP - 46
EP - 60
JO - Combinatorics Probability and Computing
JF - Combinatorics Probability and Computing
IS - 1
ER -