Ratewise-optimal non-sequential search strategies under constraints on the tests

Ahlswede R. Ratewise-optimal non-sequential search strategies under constraints on the tests. Discrete Applied Mathematics. 2008;156(9: Special Issue):1431-1443.Already in his Lectures on Search [A. Rdnyi, 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 is an element of 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 vertical bar A(i)vertical bar = log n/h (k/n) in terms of binary entropy and the upper bound m(n, k) infinity) 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 = rho n with rho infinity) 1/n log M(n, r) = 1 - h(rho). However, it must be emphasized that the methods of prove for our two upper bounds are quite different. (C) 2007 Elsevier B.V. All rights reserved