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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.

Suppose a binary string x = x1 . . . xn is being broadcast repeatedly over a faulty communication channel. Each time, the channel delivers a fixed number m of the digits (m < n) with the lost digits chosen uniformly at random and the order of the surviving digits preserved. How large does m have to be to reconstruct the message?

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Indistinguishable Sceneries on the Boolean Hypercube

A Binary Deletion Channel With a Fixed Number of Deletions

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2 results

Indistinguishable Sceneries on the Boolean Hypercube

A Binary Deletion Channel With a Fixed Number of Deletions