On the covering of vertices for fault diagnosis in hypercubes
Information Processing Letters
Bibliography of publications by Rudolf Ahlswede
Information Theory, Combinatorics, and Search Theory
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Already in his Lectures on Search [A. Renyi, Lectures on the theory of search, University of North Carolina, Chapel Hill, Institute of Statistics, Mimeo Series No. 6007, 1969. [11]] Renyi suggested to consider a search problem, where an unknown x@?X={1,2,...,n} is to be found by asking for containment in a minimal number m(n,k) of subsets A"1,...,A"m with the restrictions |A"i|==logn/h(k/n) in terms of binary entropy and the upper bound m(n,k)="~m(n,pn)/logn=1/h(p). Actually this work was motivated by a more recent study of Karpovsky, Chakrabarty, Levitin and Avresky of a problem on fault diagnosis in hypercubes, which amounts to finding the minimal number M(n,r) of Hamming balls of radius r=@rn with @r=~1nlogM(n,r)=1-h(@r).However, it must be emphasized that the methods of prove for our two upper bounds are quite different.