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

Abstract

Already in his Lectures on Search [A. Rényi, 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 | ⩽ k < n / 2 for i = 1 , 2 , … , m . Katona gave in 1966 the lower bound m ( n , k ) ⩾ log n / h ( k / n ) in terms of binary entropy and the upper bound m ( n , k ) ⩽ ⌈ ( log n + 1 ) / log n / k ⌉ · n / k , which was improved by Wegener in 1979 to m ( n , k ) ⩽ ⌈ log n / log n / k ⌉ ( ⌈ n / k ⌉ - 1 ) . We prove here for k = pn that m ( n , k ) = log n + o ( log n ) / h ( p ) , that is, ratewise optimality of the entropy bound: lim n → ∞ m ( n , pn ) / log n = 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 = ρ n with ρ ⩽ 1 2 in the Hamming space H n = { 0 , 1 } n , which separate the vertices. Their bounds on M ( n , r ) are far from being optimal. We establish bounds implying lim n → ∞ 1 n log M ( n , r ) = 1 - h ( ρ ) . However, it must be emphasized that the methods of prove for our two upper bounds are quite different.